Skip to main content

Full text of "History of formal logic"

See other formats


UNIVERSITY 
OF FLORIDA 
LIBRARIES 




A HISTORY OF FORMAL LOGIC • BOCHENSKI 



Digitized by the Internet Archive 

in 2011 with funding from 

Lyrasis Members and Sloan Foundation 



http://www.archive.org/details/historyofformallOOboch 



A HISTORY 
OF FORMAL LOGIC 



BY 

I. M. BOCHENSKI 

TRANSLATED AND EDITED BY 
IVO THOMAS 



UNIVERSITY OF NOTRE DAME PRESS . 1961 



A HISTORY OF FORMAL LOGIC 

is a revised translation by Ivo Thomas of the German edition, Formale Logik, 

by J. M. Bochehski, published and copyrighted by Verlag Karl Alber, 

Freiburg/Miinchen, in 1956 



© 1961 by 

University of Notre Dame Press, Notre Dame, Indiana 

Library of Congress Catalog Card Number 58 — 14183 

Printed in the U. S.A. 



PREFACE TO THE GERMAN EDITION 



This history of the problems of formal logic, which we believe 
to be the first comprehensive one, has grown only in small part from 
the author's own researches. Its writing has been made possible 
by a small group of logicians and historians of logic, those above 
all of the schools of Warsaw and Munster. It is the result of their 
labours that the work chiefly presents, and the author offers them 
his thanks, especially to the founders Jan Lukasiewicz and Heinrich 
Scholz. 

A whole series of scholars has been exceptionally obliging in 
giving help with the compilation. Professors E. W. Beth (Amster- 
dam), Ph. Boehner O.F.M. (St. Bonaventure, N.Y.), A. Church 
(Princeton), 0. Gigon (Bern), D. Ingalls (Harvard), J. Lukasiewicz 
(Dublin), B. Mates (Berkeley, California), E. Moody (Columbia 
University, New York), M. Morard O.P. (Fribourg), C. Regamey 
(Fribourg/Lausanne) and I. Thomas O.P. (Blackfriars, Oxford) have 
been kind enough to read various parts of the manuscript and 
communicate to me many valuable remarks, corrections and addi- 
tions. Thanks to them I have been able to remove various inexacti- 
tudes and significantly improve the content. Of course they bear no 
responsibility for the text in its final state. 

The author is further indebted for important references and 
information to Mile. M. T. d'Alverny, Reader of the Department of 
Manuscripts of the Bibliotheque Nationale in Paris, Dr. J. Yajda 
of the Centre National de la Recherche Scientifique in Paris, Pro- 
fessors L. Minio-Paluello (Oxford), S. Hulsewe (Leiden), H. Hermes 
and H. Scholz (Munster i. W.), R. Feys (Louvain) and A. Badawi 
(Fuad University, Cairo). Dr. A. Menne has been kind enough to 
read the proofs and make a number of suggestions. 

My assistant, Dr. Thomas Raber, has proved a real collaborator 
throughout. In particular, I could probably not have achieved the 
translation of the texts into German without his help. He has 
also been especially helpful in the compilation of the Bibliography 
and the preparation of the manuscript for press. 



A HISTORY OF FORMAL LOGIC 

In the course of my researches I have enjoyed the help of several 
European libraries. I should like to name here those in Amsterdam 
(University Library), Basel (University Library), Gottingen (Nie- 
dersachsische Landes- und Universitatsbibliothek), Kolmar (Stadt- 
bibliothek), London (British Museum and India Office Library), 
Munich (Bayerische Staatsbibliothek), Oxford (Bodleian Library) 
and Paris (Bibliotheque Nationale); above all the Kern-Institut in 
Leiden and the institutes f or mathematical logic in Louvain and 
Munster which showed me notable hospitality. Finally, last but 
not least, the Cantonal and University Library of Fribourg, where 
the staff has made really extraordinary efforts on my behalf. 

The completion of my inquiries and the composition of this book 
was made materially possible by a generous grant from the Swiss 
national fund for the advancement of scientific research. This 
enabled me to employ an assistant and defray the costs of several 
journeys, of microfilms etc. My best thanks are due to the adminis- 
trators of the fund, as to all who have helped me in the work. 

Since the manuscript was completed, Fr. Ph. Boehner O.F.M. 
and Dr Richard Brodfiihrer, editor of the series 'Orbis Academicus', 
have died. Both are to be remembered with gratitude. 

I. M. BOCHENSKI 



VI 



PREFACE TO THE ENGLISH EDITION 



A. GENERAL 

In this edition of the most considerable history of formal logic 
yet published, the opportunity has of course been taken to make 
some adjustments seen to be necessary in the original, with the 
author's full concurrence. Only in § 36, however, has the numeration 
of cited passages been altered owing to the introduction of new 
matter. Those changes are as follows: 

German edition English edition 

36.13 36.17 

36.14 36.18 
36.15—17 36.21—23 

Other alterations that may be noted are: the closing paragraphs of 
§ 15 have been more accurately suited to the group of syllogisms 
which they concern; 16.17 appears here as a principle rejected, not 
accepted, by Aristotle; 27.28 has been re-interpreted and the 
lengthy citation dropped; a new sub-section, on the beginnings of 
combinatory logic, has been added to § 49. A few further items have 
been added to the Bibliography. On grounds of economy this last 
has been reproduced photographically; probably such German 
remarks and abbreviations as it contains will not much inconven- 
ience its users. 

A word needs to be said about § 5 A, 'Technical Expressions', 
which has naturally had to be largely re-written. In the orginal. 
the author expressed his intention of using 'Aussage' as a name for 
sentential expressions, and so for certain dispositions of black ink, 
or bundles of sound-waves. So understood, the word could be 
treated as generally synonymous w r ith the Scholastic 'proposilio' 
and the English 'sentence' when used in an equally technical sense, 
and was deemed a tolerable translation of Russell's 'proposition' the 
reference of which is often ambiguous. But these equations evidently 
cannot be maintained here; for one thing, they would warrant the 
change of 'proposition' to 'sentence' throughout quotations from 



vn 



A HISTORY OF FORMAL LOGIC 

Russell; secondly we prefer, with A. Church (1.01), to speak of 
'propositional logic' rather than 'sentential logic' ; and thirdly, 
one risks actual falsification of one's material by imposing on it a 
grid of sharp distinctions which for the most part belongs to a 
later period than anything here treated. As noted in the body of 
the work, the Stoics and Frege were alone in making the distinction 
between sign and significate as sharply as is now customary. So we 
have normally used 'proposition' where the author used 'Aussage' - 
always, indeed, when the Scholastic 'propositio' needs translation. 
In Part V usage is of course largely conditioned by the fact that so 
many citations now appear in their original language. 

B. ABELARD 

As to the contents, an evident lacuna is the absence of any texts 
from the 12th century a.d., and the author himself has suggested 
that 30.03 is quite insufficient reference to Peter Abelard, described 
in an epitaph as 'the Aristotle of our time, the equal or superior of 
all logicians there have been' (1.02), and in similar words by John 
of Salisbury (1.03). We propose only to elaborate that single reference, 
by way of giving a taste of this twelfth century logic, closely based 
on Boethius in its past, growing in an atmosphere of keen discus- 
sion in its present, evidently holding the seeds of later Scholastic 
developments as exemplified in this book. 

Abelard's consequences are certainly not fully emancipated from 
the logic of terms (cf. 30.03) yet he realizes that propositions are 
always involved. His explanation of 'consequential' may first be 
noted : 

1.04 A hypothetical proposition is called a 'consequence' 
after its consequent, and a 'conditional' after its condition. 

Speaking later of a form of the laws of transposition (43.33) he 
says: 

1.05 My opinion is that while the force of the inference lies 
in the terms, yet the whole proposition is to be denied. . . . 
Rightly the whole sequent and antecedent proposition is to 
be denied, since the inference lies between the entire proposi- 
tions, though the force of the inference depends on the 
terms. ... So that the hypothetical proposition is rightly 
said not to be composed of simple terms, but to be conjoined 
from several propositions, inasmuch as it propounds that what 
the sequent proposition manifests, follows from what the 
precedent (manifests). So that the denial is not to be effected 
according to the terms alone, but according to the entire 
propositions between which the relation of consequence is 
propounded. 

viii 



ABELARD 

Consequences themselves are distinguished from their metalogical 
formulations (cf. the commentary preceding 31.14), the latter being 

called 'maximae propositiones' and defined thus: 

1.06 That proposition which contains the sense of many 
consequences and manifests the manner of proof common to 
their determining features (differentiae) according to the 
force of their relationship, is called a 'maximal proposition 1 . 
E.g. along with these consequences: 'if it is man, it is animal', 
'if it is pearl, it is stone', 'if it is rose, it is flower', 'if it is 
redness, it is colour' etc., in which species precede genera, 
a maximal proposition such as the following is adduced : of 
whatever the species is predicated, the genus too (is predi- 
cated). . . . This maximal proposition contains and expresses 
the sense of all such consequences and manifests the way of 
yielding inference common to the antecedents. 

There is a rich store of maxims in Abelard, but it is not always 
easy to see whether they belong to the logic of terms or propositions. 
This ambiguity has been noted with reference to Kilwardby (cf. 
§ 31 , B ) where one might be tempted to think that it was unconscious. 
But the terminology is not subject to direct attention in Kilwardby; 
in Abelard it is, and the ambiguity is noted and accepted. The 
following passage may need apology for its length, but not for its 
great interest in respect of terminology, semantic considerations 
(on which we cannot here delay), maxims both of validity and 
invalidity, and the reduction of some of them to others, 1.06 and 
1.07 are enough to establish the basis and essentials of § 31 firmly 
in the 12th century. 

1.07 'Antecedent' and 'consequent' are sometimes used to 
designate complete enunciations as when in the consequence : 
if Socrates is man, Socrates is animal, we say that the first 
categorical is antecedent to the second; sometimes in the 
designation of simple terms (dictio) or what they signify, as 
when we say in regard to the same consequence that the 
species is antecedent to the genus, i.e. 'man' to 'animal', the 
nature or relationship provides inferential force. . . . But 
whether we take 'antecedent' and 'consequent' for simple 
terms or complete enunciations, we can call them the parts of 
hypothetical enunciations, i.e. the parts of which the conse- 
quences are composed and of which they consist, not parts 
of which they treat. For we cannot accept as true this conse- 
quence: if he is man, he is animal, if it treats of utterances 
(vocibus) be they terms or propositions. For it is false that if 

ix 



A HISTORY OF FORMAL LOGIC 

the utterance 'man' exists, there should also be the utterance 
'animal'; and similarly in the case of enunciations or their 
concepts (intelledibus). For it is not necessary that he who has 
a concept generated by the precedent proposition should also 
have one generated from the consequent. For no diverse con- 
cepts are so akin that one must be possessed along with the 
other; indeed everyone's own experience will convince him 
that his soul does not retain diverse concepts and will find 
that it is totally occupied with each single concept while he 
has it. But if someone were to grant that the essences of 
concepts follow on one another like the essences of the 
things from which the concepts are gained, he would have 
to concede that every knower has an infinity of concepts 
since every proposition has innumerable consequences. 
Further, whether we treat of enunciations or of their concepts, 
we have to use their names in a consequence; but if 'man' or 
'animal' are taken as names either of enunciations or concepts, 
'if there is man there is animal' cannot at all be a consequence, 
being composed entirely of terms, as much as to say: 'if man 
animal' ; indeed as a statement it is quite imperfect. To keep, 
therefore, a genuine relation of consequence we must concede 
that it is things which are being treated of, and accept the 
rules of antecedent and consequent as given in the nature 
of things. These rules are as follows : 

(1) on the antecedent being posited, the consequent is posited; 

(2) on the consequent being destroyed, the antecedent is 
destroyed, thus: 

'if there is man there is animal', 'if there is not animal there 
is not man'; 

(3) neither if the antecedent is posited, is the consequent 
destroyed, 

(4) nor if the antecedent is destroyed need the consequent 
be destroyed 

(5) or posited, just as 

(6) neither if the consequent is destroyed is the antecedent 
posited, 

(7) nor if the same (the consequent) is posited is it (the ante- 
cedent) either posited 

(8) or removed. 

Since the last ( (6)-(8) ) are equivalent to the former ( (3)-(5) ) 
as also their affirmatives are mutually equivalent, the two 
sets must be simultaneously true or false. The two first rules 



ABELARD 

are also in complete mutual agreement and can be derived 
from one another, e.g. if it is conceded : if there is man there 
is animal, it must also be conceded: 'if there is not animal 
there is not man, and conversely. 

When the first is true, the second will be proved true as 
follows, by inducing an impossibility. Let us posit this as 
true: if there is man there is animal, and doubt about this: 
if there is not animal there is not man, i.e. whether 'animal' 
negated negates 'man'. We shall confirm this in the following 
way. Either 'animal' negated negates 'man' or negated it 
admits 'man', so that it may* happen that when 'animal' is 
denied of something man may exist in that thing. Suppose it 
be conceded that when 'animal' is denied, man may persist; 
yet it was formerly conceded that 'man' necessarily requires 
'animal', viz. in the consequence: if there is man there is 
animal. And so it is contingent that what is not animal, be 
animal; for what the antecedent admits, the consequent 
admits. . . . But this is impossible. . . . 

Quite definitely propositional are the rules: 

1.08 Whatever follows from the consequent (follows) also 
from the antecedent; 

Whatever implies the antecedent (implies) also the conse- 
quent; 

used in the derivation of categorical syllogisms. 

While it is clear that the primary source of all this doctrine of 
consequence is the De Differentiis Topicis of Boethius, we can also 
see the germ of later developments in Abelard's realization that some 
are deducible from others (1.07, 1.09), and his examinations of some 
that he finds doubtful (1.10). 

It is noteworthy that categorical syllogisms are presented entirely 
by means of concrete instances and metalogical rules (regulae) — 
which are not reckoned as maxims since the inferential (or impli- 
cative) 'force' of the premisses is derived entirely from the disposi- 
tion of the terms, is, in Abelard's terminology, 'complexional', a 
term preserved in Kilwardby. Variables of the object-language are 
nowhere used. Indeed, except in expositions of the Boethian 
hypothetical syllogisms, the only place we find variables in Abelard 
is a passage where he introduces a simple lettered diagram to help 
the intuition in an original argument: 

1.11 If a genus was always to be divided into proximate 
species or proximate differences, every division of a genus 

* Emending De Rijk's possint to possit 

xi 



A HISTORY OF FORMAL LOGIC 

would be dichotomous - which was Boethius's view. . . . But 
I remember having an objection to this on the score of (the 
predicament of) relation. . . . This will be more easily seen if 
we designate the members of the predicament by letters and 
distinguish its arrangement by a figure like this. 

Relation 



D F G L 

If now C and D were mutually related on the one hand, 
B and C on the other, since B is prior to its species Z), while 
D is together with (simul) its relative C, B would precede C; 
so that B would precede both its species and its relative; 
hence also itself. 

There follow two more arguments to show that the system suppos- 
ed figured stands or falls entire with any one of its parts. 

There is no suggestion in Abelard that syncategorematica, 
important for later theory of consequences, are a primary concern 
of logic, the purpose of which he states as follows: 

1.12 Logic is not a science of using or composing arguments, 
but of discerning and estimating them rightly, why some are 
valid, others invalid. 

But he is puzzled about the signification, if any, of syncategore- 
matica, and refers to various contemporary views: 

1.13 Conjunctions and prepositions ought to have some 
signification of their own. . . . What concepts are designated by 
expressions of this kind, it is not easy to say. . . . Some think 
that such expressions have sense but no reference (solos 
intelledus generare, nullamque rem subiedam habere) as they 
grant also to be the case with propositions. . . . There are also 
some who make out that logicians have quite removed such 
expressions from the class of significant ones. . . . The opinion 
I favour is that of the grammarians who make contributions 
to logic, that we should admit them as significant, but should 
say that their significance lies in their determining certain 
properties of the references (res) of the words governed by the 
prepositions. . . . Conjunctions too, as indicating conjunction 
of things, determine a property in their regard, e.g. when I 
say: 'a man and a horse runs', by the conjunction 'and' I 
unite them in runing, and at the same time indicate that by 
the 'and'. 



Xll 



ABELARD 

The emergence of a logic of propositions from one of terms is 
exemplified in the rather sophisticated, and disputed, distinction 
between propositional and term connectives : 

1.14 It is to be remarked that while disjunctive connectives 
are applied to the terms both of categorical and hypothetical 
propositions, they seem to have a different sense in each. . . . 
But some allow no difference . . . saying that there is the 
same proposition when it is said: 'Socrates is healthy or (we/) 
sick' and when it is said: 'Either Socrates is healthy or (aut) 
he is sick', reckoning every disjunctive as hypothetical. 

Again of temporal propositions, compounded by means of 'when', 
which Abelard treats as conjunctives, he says: 

1.15 It is evident in temporals that we should not estimate 
the relation of consequence according to any force in the 
relationship of terms, . . . but only in the mutual accompani- 
ment (of the components). 

And again, with the addition of truth-conditions: 

1.16 In these (temporals) in which the relation of conse- 
quence is to be reckoned nothing else than coincidence in 
time . . . provided the members are true, people concede that 
the consequence is true, and otherwise false. 

Some 'rules' follow for the construction of consequences on this 
basis. Among them the following deserves special notice : 

1.17 Of whatever (hypotheticals) the antecedents are 
concomitant, the consequents too (are concomitant), thus: 
if when he is a man he is a doctor, when he is an animal he is 
an artificer. 

This is Leibniz's praeclarum Iheorema (cf. 43.37) in essentials, 
though it seems impossible to say whether Abelard envisaged it in 
its Leibnizian classical or its Russellian propositional form. He 
explains it indeed by saying that 'as "man" is necessarily antecedent 
to "animal", so "doctor" is to "artificer"', yet he clearly thinks of 
it as compounded of propositions. The fact is that the two kinds of 
logic were not yet perfectly clearly distinguished. A further indication 
that the full generality of propositional logic had not yet been achie- 
ved, though, as we have seen, it was already in the making, is that 
while Abelard gives us as a consequence: if he is man and stone, he 
is animal, he does not rise to: if he is man and stone, he is man. 

There is evidently a vast deal more to be said both about the 
prowess and the limitations of this logician, both on this and other 
subjects, but we already exceed the limits of discussion proper to 
this history. It is, however, certain that the serious beginnings of 
Scholastic logic must be looked for in the 12th century. 

IVO THOMAS 



xm 



ACKNOWLEDGEMENTS 



Grateful acknowledgements are due to Bertrand Russell for 
permission to quote from his The Principles of Mathematics 
(London, 1903), and to the respective publishers for permission 
to use passages from: The Dialogues of Plato, translated in 
English by B. Jowett (Oxford University Press); The Works 
of Aristotle, translated into English under the editorship of 
J. A. Smith andW. D.Ross (Oxford University Press) ; Principia 
Mathematica, by A. N. Whitehead and Bertrand Russell (Cam- 
bridge University Press, 1925-27) ; Traciatus Logico-Philo- 
sophicus, by L.Wittgenstein (Routledge and Kegan Paul, 1922) ; 
Translations from the Philosophical Writings of Gottlob Frege, 
by Peter Geach and Max Black (Basil Blackwell, Oxford, 1952) ; 
to The Belknap Press of Harvard University Press, for quota- 
tions from The Collected Papers of Charles Sanders Peirce, edited 
by Charles Hartshorne and Paul Weiss (copyright 1933 by the 
President and Fellows of Harvard University). 



CONTENTS 



Preface to the German edition v 

Translator's preface to the English edition 

A. General vii 

B. Abelard viii 



PART I 
Introduction 

§ 1. The concept of formal logic 2 

§ 2. On the history of the history of logic 4 

A. The beginnings 4 

B. Prejudices 4 

1. Thomas Reid 5 

2. Kant 6 

3. Prantl 6 

4. After Prantl 8 

C. Research in the 20th century 9 

§ 3. The evolution of formal logic 10 

A. Concerning the geography and chrorretogy of logic ... 10 

B. How logic evolved 12 

C. The varieties of logic 12 

D. The unity of logic 14 

E. The problem of progress 15 

§ 4, Method and plan 18 

A. History of problems, and documentation 18 

B. Plan of the work 18 

C. Character of the contents 19 

§ 5. Terminology 20 

A. Technical expressions 20 

B. Concerning mathematico-logical symbolism .... 22 

C. Typographical conventions 23 

PART II 
The Greek Variety of Logic 

§ 6. Introduction to Greek logic 26 

A. Logicians in chronological order 26 

B. Periods '27 

C. State of research 27 



XV 



CONTENTS 

I. The precursors 

§ 7. The beginnings 29 

A. Texts 29 

B. Significance 31 

§ 8. Plato 33 

A. Concept of logic 33 

B. Approaches to logical formulae 34 

C. Diaeresis 35 

II. Aristotle 

§ 9. The work of Aristotle and the problems of its literary history . . 40 

A. Works 40 

B. Problems 40 

1. Authenticity 40 

2. Character 41 

3. Chronology 41 

§ 10. Concept of logic. Semiotic 44 

A. Name and place of logic 44 

B. The subject-matter of logic 44 

C. Syntax 46 

D. Semantics 47 

§ 11. The topics 49 

A. Subject and purpose 49 

B. Predicables 51 

C. Categories 53 

D. Sophistic 54 

§ 12. Theory of opposition; principle of contradiction; principle of tertium 

exclusum 57 

A. Theory of opposition 57 

B. Obversion 59 

C. The principle of contradiction 60 

D. The principle of tertium exclusum 62 

§ 13. Assertoric syllogistic 63 

A. Text 64 

B. Interpretation 66 

C. Structure of the syllogism 69 

D. The figures and further syllogisms 70 

§ 14. Axiomatization of the syllogistic. Further laws 72 

A. Axiomatic theory of the system 72 

B. Systems of syllogistic 75 

C. Direct proof 76 

D. Indirect proof 77 

E. Dictum de omni et nullo 79 

F. Beginnings of a metalogical system 80 

G. The inventio medii 80 

§ 15. Modal logic 81 

A. Modalities 81 

B. Structure of modal sentences 83 

C. Negation and conversion 84 

D. Syllogisms 85 

§ 16. Non-analytic laws and rules 88 

A. Two kinds of inference 89 

B. Laws of class- and predicate-logic 91 

C. Theory of identity 92 

D. Syllogisms from hypotheses 93 

E. Laws of the logic of relations 95 

F. Propositional rules and laws 97 

Summary 98 

xvi 



CONTENTS 

§ 17. Theophrastus 99 

A. Development and alteration of various doctrines . . . 99 

B. Modal logic 101 

C. Hypothetical syllogisms 103 

III. The Megarian-Stoic School 

§ 18. Historical survey 105 

A. Thinkers and schools 105 

B. Problems of literary history 107 

C. Origin and nature . 108 

§ 19. Concept of logic. Semiotics. Modalities 109 

A. Logic 109 

B. Lecta 110 

C. Syntax Ill 

D. Doctrine of categories 113 

E. Truth 114 

F. Modalities 114 

§ 20. Proposilional functors 115 

A. Negation 116 

B. Implication 116 

1. Philonian implication 117 

2. Diodorean implication 117 

3. 'Connexive' implication 118 

4. 'Inclusive' implication 119 

C. Disjunction 119 

1. Complete disjunction 119 

2. Incomplete disjunction 120 

D. Conjunction 121 

E. Equivalence 121 

F. Other functors 121 

§ 21. Arguments and schemes of inference 122 

A. Conclusive, true and demonstrative arguments . . . . 122 

B. Non-syllogistic arguments 124 

C. Further kinds of argument 124 

D. Schemes of inference 125 

§ 22. Axiomatization. Compound arguments 126 

A. The indemonstrables 126 

B. Metatheorems 127 

C. Derivation of compound arguments 128 

D. Further derived arguments 130 

§ 23. The liar 130 

A. History 131 

B. Formulation 131 

C. Efforts at solution 132 

IV. The Close of Antiquity 

§ 24. Period of commentaries and handbooks 134 

A. Characteristics and historical survey 134 

B. The tree of Porphyry 135 

C. Extension of logical technique 136 

D. Fresh division of implication 137 

E. Boethius's hypothetical syllogisms 139 

F. Alterations and development of the categorical syllogistic . 140 

G. The supposedly fourth figure 141 

H. Pons asinorum 143 

I. Anticipation of the logic of relations 144 

Summary 144 

xvii 



CONTENTS 

PART III 

The Scholastic Variety of Logic 

§ 25. Introduction to scholastic logic 148 

A. State of research 148 

B. Provisional periods 148 

C. The problem of sources 150 

D. Logic and the schools 150 

E. Method 151 

F. Characteristics 152 

I. Semiotic foundations 

§ 26. Subject-matter of logic 153 

A. Basic notions of semiotics 153 

B. Logic as a theory of second intentions 154 

C. Formal logic as a theory of syncategorematic expressions . 156 

D. Content of the works 159 

§ 27. Supposition 162 

A. Concept of supposition 163 

B. Material and formal supposition 164 

C. Simple supposition 168 

D. Personal supposition 171 

E. Interpretation in modern terms 173 

§ 28. Ampliation, appellation, analogy 173 

A. Ampliation 173 

B. Appellation 175 

C. Analogy 177 

§ 29. Structure and sense of propositions 180 

A. Division of propositions 180 

B. Analysis of propositions 180 

C. Analysis of modal propositions: dictum and modus . . . 182 

D. Composite and divided senses 184 

E. Reference of propositions 187 

II. Propositional Logic 

§ 30. Notion and division of consequences 189 

A. Historical survey 189 

B. Definition of consequence 190 

C. Division of consequences 191 

D. Meaning of implication 195 

E. Disjunction 197 

§ 31. Propositional consequences 198 

A. Hypothetical propositions 198 

B. Kilwardby 198 

C. Albert of Saxony 199 

D. Paul of Venice 205 

E. Rules of consequences ut nunc 208 

III. Logic of Terms 

§ 32. Assertoric syllogistic 210 

A. Early mnemonics 210 

B. Barbara-Celarent 211 

C. Barbara-Celaront 215 

D. The fourth figure 216 

1. Among the Latins 216 

2. In Albalag 217 

E. Combinatorial method 219 

F. Inventio medii, pons asinorum 219 

xviii 



CONTENTS 

G. The problem of the null class 221 

1. St. Vincent Ferrer 221 

2. Paul of Venice 223 

3. John of St. Thomas 223 

§ 33. Modal syllogistic 224 

A. Albert the Great 224 

B. Pseudo-Scotus 225 

C. Ockham 227 

D. Logic of propositions in future and past tenses .... 230 
§ 34. Other formulae 231 

A. Syllogisms with singular terms 232 

B. Analysis of 'every' and 'some' 234 

C. Exponible propositions 234 

D. Oblique syllogisms 236 

§ 35. Antinomies 237 

A. Development 237 

B. Formulation 239 

1. The liar 239 

2. Other antinomies 240 

C. Solutions 241 

1. The first twelve solutions 241 

2. The thirteenth solution 244 

3. The fourteenth solution 246 

4. Preliminaries to the solution of Paul of Venice . . . 247 

5. The solution of Paul of Venice 249 

Summary 251 

PART IV 
Transitional Period 

§ 36. The 'classical' logic 254 

A. Humanism 254 

B. Content 256 

C. Psychologism 257 

D. Leibniz 258 

E. Comprehension and extension 258 

F. The fourth figure and subaltern moods 259 

G. Syllogistic diagrams 260 

H. Quantification of the predicate 262 

PART V 
The Mathematical Variety of Logic 

I. General Foundations 

§37. Introduction to mathematical logic 266 

A. Characteristics 266 

B. Chronological sequence 267 

C. Frege 268 

D. Periods 269 

E. State of research 270 

F. Method 271 

§ 38. Methods of mathematical logic 272 

A. Logical calculus 272 

1. Lull 272 

2. Hobbes 273 

3. Leibniz 274 



XIX 



CONTENTS 

4. Lambert 276 

5. Gergonne 277 

6. Boole 278 

7. Peirce 279 

B. Theory of proof 280 

1. Bolzano 280 

2. Frege 282 

C. Metalogic 284 

§39. The concept of logic 286 

A. The logistic position 287 

1. Frege: semantics 287 

2. Frege: logic and mathematics 289 

3. Russell 290 

4. Frege: number 291 

B. Formalism 292 

C. Intuitionism 293 

II. The First Period 

§ 40. The Boolean calculus 296 

A. De Morgan 296 

B. Boole 298 

1. Symbolism and basic concepts 298 

2. Applications 301 

C. The logical sum 302 

D. Inclusion 303 

E. Peano 306 

III. Propositional Logic 

§ 41. Propositional logic: basic concepts and symbolism .... 307 

A. Boole 307 

B. McColl 309 

C. Frege 310 

1. Content and judgment 310 

2. Implication 311 

D. Peirce 313 

E. Applications of his symbolism by Frege 314 

F. Negation and sum in Frege 316 

G. Peano's symbolism for propositional logic 317 

H. Later development of symbolism for propositional logic . . 318 

§ 42. Function, variable, truth-value 319 

A. Logical form 320 

B. Concept of function: Frege 320 

C. Propositional function: Russell 322 

D. Many-place functions 323 

E. The variable 325 

1. Frege 325 

2. Russell 326 

F. Truth-values 327 

G. Truth-matrices 330 

1. Peirce 330 

2. Wittgenstein 331 

H. Decision procedure of Lukasiewicz 333 

§ 43. Propositional logic as a system 335 

A. McColl 335 

B. Frege's rules of inference 337 

C. Propositional laws from the Begriffsschrift 338 

D. Whitehead and Russell 340 

1. Primitive symbols and definition 340 



CONTENTS 

2. Axioms (primitive propositions) 340 

3. Statement of proofs 341 

4. Laws 342 

E. Sheffer's functor 344 

F. Lukasiewicz's statement of proofs 345 

IV. Logic of Terms 

§ 44. Predicate logic 347 

A. Quantifiers 348 

1. Mitchell 348 

2. Peirce . 348 

3. Peano 349 

4. Frege 350 

B. Apparent variables 353 

1. Peano 353 

2. Whitehead and Russell 353 

C. Formal implication 354 

D. Laws of one-place predicates 355 

E. Laws of many-place predicates 356 

F. Identity 357 

§ 45. The logic of classes 359 

A. Individual and class. Concept of element 360 

B. Meaning and extension 360 

C. The plural article 362 

D. Definition of classes by functions 363 

E. Product and inclusion of classes 364 

§ 46. Existence 365 

A. The null class 365 

B. Null class and assertoric syllogistic 366 

C. Description 367 

1. The definite article: Frege 367 

2. Logical existence 369 

3. Description in Russell 371 

4. Symbolism 373 

(a) Peano 373 

(b) Principia 374 

V. Other Doctrines 

§ 47. Logic of relations 375 

A. Laying the foundations 375 

1. De Morgan 375 

2. Peirce 377 

3. Russell 379 

4. Principia 380 

B. Series 3S4 

1. Frege 385 

2. Principia 385 

C. Isomorphy 386 

§ 48. Antinomies and theories of types 387 

A. Historical survey 3S7 

B. The antinomies 388 

C. Anticipations of the theory of types 391 

D. The ramified theory of types 393 

E. Systematic ambiguity 396 

F. The axiom of reducibility 398 

G. Simple theory of types 399 

1. Chwistek 399 

2. Ramsey 400 



XXI 



CONTENTS 

§ 49. Some recent doctrines 402 

A. Strict implication: Lewis 403 

B. Many-valued logics: Lukasiewicz 405 

C. Godel's theorem 407 

D. Combinatory logic 411 

Summary 412 

PART VI 
The Indian Variety of Logic 

§ 50. Introduction to Indian logic 416 

A. Historical survey 416 

B. Evolution of formal logic 417 

C. State of research 419 

D. Method 420 

§ 51. The precursors 421 

A. Milinda-Panha 421 

B. Kathavatthu 421 

C. The ten-membered formula 423 

§ 52. Vaisesika- and Nyaya-sutra 425 

A. Vaisesika-sutra 425 

1. Doctrine of categories 425 

2. Inference 426 

B. Nyaya-sutra 426 

1. Text 426 

2. Vatsyayana's commentary 428 

3. Interpretation 429 

§ 53. The rise of formal logic . 431 

A. Main stages of development 431 

B. Terminology 432 

C. The three-membered syllogism 433 

D. The three-membered rule: trairupya 435 

E. Wheel of reasons: hetu-cakra 435 

F. 'Eva' 437 

G. Universal connection 437 

H. Final form of the doctrine 439 

§ 54. Some other logical doctrines 440 

A. Apoha 441 

B. Definitions of vyapti 441 

C. Some basic concepts 442 

D. The law of double negation 444 

E. Relation logic, definition of number 445 

Summary 446 



REFERENCES, BIBLIOGRAPHY, INDICES 

I. References 451 

II. Bibliography 460 

III. Indices 535 

1. Index of names 535 

2. Index of logical symbols 541 

3. Index of mnemonics 544 

4. Subject index 545 

Plate I opposite p. 220 

Plate II opposite p. 260 

Plate III opposite p. 274 

Plate IV opposite p. 316 



XX11 



PART I 

Introduction 



1 Bochenski, Formal Logic 



§1. THE CONCEPT OF FORMAL LOGIC 

Preliminary definition of the subject matter of the history of 
logic is hard to come by. For apart from 'philosophy' there is perhaps 
no name of a branch of knowledge that has been given so many 
meanings as 'logic'. Sometimes the whole of philosophy, and even 
knowledge in general, has been thus named, from metaphysics on 
the one hand, cf. Hegel, to aesthetics ('logic of beauty') on the other, 
with psychology, epistemology, mathematics etc. in between. With 
such a wide choice it is quite impossible to include in a history of 
logical problems all that has been termed 'logic' in the course of 
western thought. To do so would practically involve writing a 
general history of philosophy. 

But it does not follow that the use of the name 'logic' must be 
quite arbitrary, for history provides several clues to guide a choice 
between its many meanings. This choice can be arrived at by the 
following stages. 

1. First let us discard whatever most authors either expressly 
ascribe to some other discipline, or call 'logic' with the addition of 
an adjective, as for example epistemology, transcendental logic, 
ontology etc. 

2. When we examine what remains, we find that there is one 
thinker who so distinctly marked out the basic problems of this 
residual domain that all later western inquirers trace their descent 
from him: Aristotle. Admittedly, in the course of centuries very 
many of these inquirers - among them even his principal pupil and 
successor Theophrastus - have altered Aristotelian positions and 
replaced them with others. But the essential problematic of their work 
was, so far as we know, in constant dependence in one way or another 
on that of Aristotle's Organon. Consequently we shall denote as 
'logic' primarily those problems which have developed from that 
problematic. 

3. When we come to the post-Aristotelian history of logic, we can 
easily see that one part of the Organon has exercised the most 
decisive influence, namely the Prior Analytics. At some periods 
other parts too, such as the Topics or the Posterior Analytics, have 
indeed been keenly investigated and developed. But it is generally 
true of all periods marked by an active interest in the Organon that 
the problems mainly discussed are of the kind already to hand in 
the Prior Analytics. So the third step brings us to the point of 
describing as 'logic' in the stricter sense that kind of problematic 
presented in the Prior Analytics. 

4. The Prior Analytics treats of the so-called syllogism, this being 
defined as aXoyo<;in which if something is posited, something else 
necessarily follows. Moreover such Xoyot, are there treated as 
formulas which exhibit variables in place of words with constant 



CONCEPT OF FORMAL LOGIC 

meaning; an example is 'B belongs to all A'. The problem evidently, 
though not explicitly, presented by Aristotle in this epoch-making 
work, could be formulated as follows. What formulas of the prescri- 
bed type, when their variables are replaced by constants, yield 

conditional statements such that when the antecedent is accepted, 
the consequent must be admitted? Such formulas are called 'logical 
sentences'. We shall accordingly treat sentences of this kind as a 
principal subject of logic. 

5. Some logicians have limited themselves to the discovery, 
examination and systematic ordering of logical theorems, e.g. many 
scholastic and mathematical logicians, as also Aristotle himself in 
the Prior Analytics. But logic so understood seems too narrowly 
conceived. For two kinds of problem naturally arise out of the 
theorems. First those about their nature - are they linguistic 
expressions, word-structures, psychical forms or functions, objective 
complexes? What does a logical law mean, what does a statement 
mean? These are problems which nowadays are dealt with in semio- 
tics. Second, problems relevant to the question how logical laws 
can be correctly applied to practical scientific thought. These were 
dealt with by Aristotle himself, principally in the Posterior Analytics, 
and nowadays are the concern of general methodology. So semiotic 
and methodological problems are closely connected with logic; 
in practice they are always based on semiotics and completed in 
methodology. What remains over and above these two disciplines 
we shall call formal logic. 

6. A complete history of the problems of logic must then have 
formal logic at its centre, but treat also of the development of 
problems of semiotics and methodology. Before all else it must put 
the question : what problems were in the past posited with reference 
to the formulation, assessment and systematization of the laws of 
formal logic? Beyond that it must look for the sense in which these 
problems were understood by the various logicians of the past, and 
also attempt to answer the question of the application of these 
laws in scientific practice. We have now delimited our subject, and 
done so, as we think, in accordance with historical evidence. 

But such a program has proved to be beyond accomplishment. 
Not only is our present knowledge of seim'otic and methodological 
questions in the most important periods too fragmentary, but even 
where the material is sufficiently available, a thorough treatment 
would lead too far afield. Accordingly we have resolved to limit 
ourselves in the main to matters of purely formal logic, giving only 
incidental consideration to points" from the other domains. 

Thus the subject of this work is constituted by those problems 
which are relevant to the structure, interconnection and truth of 
sentences of formal logic (similar to the Aristotelian syllogism). 
Does it or does it not follow? And, why? How can one prove the 



A HISTORY OF FORMAL LOGIC 

validity of this or that sentence of formal logic? How define one 
or another logical constant, e.g. 'or', 'and', 'if--then', 'every' etc. 
Those are the questions of which the history will here be considered. 



§2. ON THE HISTORY OF THE HISTORY 
OF LOGIC 

A. THE BEGINNINGS 

The first efforts to write a history of logic are to be found among 
the humanists, and perhaps Petrus Ramus may here be counted as 
the first historian. In his Scholarum dialeclicarum libri XX we find 
some thirty long colums allotted to this history. To be sure, Ramus's 
imagination far outruns his logic: he speaks of a logica Patrum 
in which Noah and Prometheus figure as the first logicians, 
then of a logica mathematicorum which alludes to the Pythagoreans. 
There follow a logica physicorum (Zeno of Elea, Hippocrates, Demo- 
critus etc.), the logica Socratis, Pyrrhonis el Epicrelici (sicl), the 
logica Anlislheniorum el Stoicorum (here the Megarians too are 
named, among others Diodorus Cronus) and the logica Academiorum. 
Only then comes the logica Peripaleticorum where Ramus mentions 
what he calls the Arislotelis bibliotheca i.e. the Organon (which 
according to him, as in our own time according to P. Zurcher. S.J., is 
not by Aristotle), and finally the logica Aristoteleorum inter prelum el 
praecipue Galeni (2.01). 

This book was written in the middle of the 16th century. Some 
fifty years later we find a less comprehensive but more scientific 
attempt by B. Keckermann. His work (2.02) is still valuable, parti- 
cularly for a large collection of accurately dated titles. It remains an 
important foundation for the study of lGth century logic. But its 
judgments are not much more reliable than those of Ramus. Kecker- 
mann seems to have given only a cursory reading to most of the 
logicians he cites, Hospinianus (2.03) for example. The book is 
indeed more of a bibliography than a history of logic. 

B. PREJUDICES 

For all his faults, Ramus was a logician; Keckermann too had 
some knowledge of the subject. The same can seldom be said of their 
successors until Bolzano, Peirce and Peano. Most historians of logic 
in the 17th, 18th and 19th centuries treat of ontological, epistemolo- 
gical and psychological problems rather than of logical ones. Further- 
more, everything in this period, with few exceptions, is so condi- 
tioned by the then prevailing prejudices that we may count the 
whole period as part of the pre-history of our science. 



HISTORY OF THE HISTORY OF LOGIC 

These prejudices are essentially three: 

1. First, everyone was convinced that formalism has very little 
to do with genuine logic. Hence investigations of formal logic either 
passed unnoticed or were contemptuously treated as quite subsidiary. 

2. Second, and in part because of the prejudice already mentioned, 
the scholastic period was treated as a media tempestas, a 'dark 
middle age' altogether lacking in science. But as the Scholastics 
were in possession of a highly developed formal logic, people sought 
in history either for quite different 'logics' from that of Aristotle 
(not only those of Noe and Epictetus, as Ramus had done, but even, 
after his time, that of Ramus himself), or at least for a supposedly 
better interpretation of him, which put the whole investigation on the 
wrong track. 

3. Finally, of equal influence was a strange belief in the linear 
development of every science, logic included. Hence there was a 
permanent inclination to rank inferior 'modern' books higher than 
works of genius from older classical writers. 

1. Thomas Reid 

As an example of how history was written then, we shall cite one 
man who had the good will at least to read Aristotle, and who 
succeeded in doing so for most parts of the Organon, though he 
failed for just the most important treatises. Here are his own words 
on the subject: 

2.04 In attempting to give some account of the Analytics 
and of the Topics of Aristotle, ingenuity requires me to con- 
fess, that though I have often purposed to read the whole 
with care, and to understand what is intelligible, yet my 
courage and patience always failed before I had done. Why 
should I throw away so much time and painful attention 
upon a thing of so little real use ? If I had lived in those ages 
when the knowledge of Aristotle's Organon entitled a man to 
the highest rank in philosophy, ambition might have induced 
me to employ upon it some years' painful study; and less, I 
conceive, would not be sufficient. Such reflections as these, 
always got the better of my resolution, when the first ardour 
began to cool. All I can say is, that I have read some parts of 
the different books with care, some slightly, and some perhaps 
not at all. ... Of all reading it is the most dry and the most 
painful, employing an infinite labour of demonstration, about 
things of the most abstract nature, delivered in a laconic 
style, and often, I think, with affected obscurity; and all to 
prove general propositions, which when applied to particular 
instances appear self-evident. 



A HISTORY OF FORMAL LOGIC 

In the first place this is a really touching avowal that Reid 
lectured on the teaching of a logician whom he had not once read 
closely, and, what is much more important, that for this Scottish 
philosopher formal logic was useless, incomprehensible and tedious. 
But beyond that, the texts that seemed to him most unintelligible 
and useless are just those that every logician counts among the most 
exquisite and historically fruitful. 

Nearly all philosophers of the so-called modern period, from the 
humanists to the rise of mathematical logic, held similar views. In 
such circumstances there could be no scientific history of logic, for 
that presupposes some understanding of the science of logic. 

The attitude towards formal logic just described will be further 
illustrated in the chapter on 'classical' logic. Here we shall delay 
only on Kant, who expressed opinions directly relevant to the 
history of logic. 

2. Kanl 

Kant did not fall a victim to the first and third of the prejudices 
just mentioned. He had the insight to state that the logic of his 
time — he knew no other — was no better than that of Aristotle, and 
went on to draw the conclusion that logic had made no progress 
since him. 

2.05 That Logic has advanced in this sure course, even from 
the earliest times, is apparent from the fact that since Ari- 
stotle, it has been unable to advance a step, and thus to all 
appearance has reached its completion. For if some of the 
moderns have thought to enlarge its domain by introducing 
psychological discussions . . . metaphysical ... or anthropological 
discussions . . . this attempt, on the part of these authors, 
only shows their ignorance of the peculiar nature of logical 
science. We do not enlarge, but disfigure the sciences when we 
lose sight of their respective limits, and allow them to run into 
one another. Now logic is enclosed within limits which admit 
of perfectly clear definition; it is a science which has for its 
object nothing but the exposition and proof of the formal laws 
of all thought, whether it be a priori or empirical. . . . 

3. Prantl 

It is a remarkable fact, unique perhaps in the writing of history, 
that Carl Prantl, the first to write a comprehensive history of 
western logic (2.06), on which task he spent a lifetime, did it pre- 
cisely to prove that Kant was right, i.e. that formal logic has no 
history at all. 

His great work contains a collection of texts, often arranged from 

6 



HISTORY OF THE HISTORY OF LOGIC 

a wrong standpoint, and no longer sufficient but still indispensable. 
He is the first to take and discuss seriously all the ancient and scho- 
lastic logicians to whom he had access, though mostly in a polemi- 
cal and mistaken spirit. Hence one can say that he founded the 
history of logic and bequeathed to us a work of the highest utility. 

Yet at the same time nearly all his comments on these logicians 
are so conditioned by the prejudices we have enumerated, are written 
too with such ignorance of the problems of logic, that he cannot be 
credited with any scientific value. Prantl starts from Kant's asser- 
tion, believing as he does that whatever came after Aristotle was 
only a corruption of Aristotle's thought. To be formal in logic, is in 
his view to be unscientific. Further, his interpretations, even of 
Aristotle, instead of being based on the texts, rely only on the 
standpoint of the decadent 'modern' logic. Accordingly, for example, 
Aristotelian syllogisms are misinterpreted in the sense of Ockham, 
every formula of propositional logic is explained in the logic of 
terms, investigation of objects other than syllogistic characterized as 
'rank luxuriance', and so of course not one genuine problem of 
formal logic is mentioned. 

While this attitude by itself makes the work wholly unscientific 
and, except as a collection of texts, worthless, these characteristics 
are aggravated by a real hatred of all that Prantl, owing to his 
logical bias, considers incorrect. And this hatred is extended from 
the teachings to the teachers. Conspicuous among its victims are the 
thinkers of the Megarian, Stoic and Scholastic traditions. Ridicule, 
and even common abuse, is heaped on them by reason of just those 
passages where they develop manifestly important and fruitful 
doctrines of formal logic. 

We shall illustrate this with some passages from his Geschichte der 
Logik, few in number when compared with the many available. 

Ghrysippus, one of the greatest Stoic logicians, 'really accom- 
plished nothing new in logic, since he only repeats what the Peri- 
patetics had already made available and the peculiarities introduced 
by the Megarians. His importance consists in his sinking to handle the 
material with a deplorable degree of platitude, triviality and scho- 
lastic niggling', Chrysippus 'is a prototype of all pedantic narrow- 
mindedness' (2.07). Stoic logic is in general a 'corruption' of that 
previously attained (2.08), a 'boundless stupidity', since 'Even he 
who merely copies other people's work, thereby runs the risk of 
bringing to view only his own blunders' (2.09). The Stoic laws are 
'proofs of poverty of intellect' (2.10). And the Stoics were not only 
stupid; they were also morally bad men, because they were subtle: 
their attitude has 'not only no logical worth, but in the realm of 
ethics manifests a moment deficient in morality' (2.11). — Of 
Scholasticism Prantl says : 'A feeling of pity steals over us when we 
see how even such partialities as are possible within an extremely 



A HISTORY OF FORMAL LOGIC 

limited field of view are exploited with plodding industry even to the 
point of exhaustion, or when centuries are wasted in their fruitless 
efforts to systematize nonsense' (2.12). Consequently 'so far as 
concerns the progress of every science that can properly be termed 
philosophy, the Middle Ages must be considered as a lost millennium' 
(2.13). 

In the 13th century and later, things are no better. 'Between the 
countless authors who without a single exception subsist only on the 
goods of others, there is but one distinction to be made.. There are 
the imbeciles such as e.g. Albertus Magnus and Thomas Aquinas, 
who hastily collect ill-assorted portions of other people's wealth in a 
thoughtless passion for authority; and those others, such as e.g. 
Duns Scotus, Occam and Marsilius, who at least understand with 
more discernment how to exploit the material at hand' (2.14). 
'Albertus Magnus too . . . was a muddle-head' (2.15). To take 
Thomas Aquinas 'for a thinker in his own right' would be 'a great 
mistake' (2.16). His pretended philosophy is only 'his unintelligent 
confusion of two essentially different standpoints; since only a 
muddled mind can . . .' etc. (2.17). 

Similar judgment is passed on later scholastic logic; a chapter on 
the subject is headed 'Rankest Luxuriance' (2.18). Prantl regrets 
having to recount the views of these logicians, 'since the only alter- 
native interpretation of the facts, which would consist simply in 
saying that this whole logic is a mindless urge, would be blameworthy 
in a historian, and without sufficient proof would not gain credence' 
(2.19). 

To refute Prantl in detail would be a huge and hardly profitable 
task. It is better to disregard him entirely. He must, unhappily, be 
treated as non-existent by a modern historian of logic. Refutation is 
in any case effected by the total results of subsequent research as 
recapitulated in this book. 

4. After Prantl 

Prantl exercised a decisive influence on the writing of history 
of logic in the 19th and to some extent also in the 20th century. Till 
the rise of the new investigations deriving from circles acquainted 
with mathematical logic, Prantl's interpretations and evaluations 
were uncritically accepted almost entire. For the most part, too, the 
later historians of logic carried still further than Prantl the mingling 
of non-logical with logical questions. This can be seen in the practice 
of giving a great deal of space in their histories to thinkers who were 
not logicians, and leaving logicians more and more out of account. 

Some examples follow. F. Ueberweg, himself no mean logician, 
(he could, e.g. distinguish propositional from term-logic, a rare gift 
in the 19th century), devoted four pages of his survey of the history 
of logic (2.20) to Aristotle, two to 'the Epicureans, Stoics and 

8 



HISTORY OF THE HISTORY OF LOGIC 

Sceptics', two to the whole of Scholasticism - but fifty-five to the 
utterly barren period stretching from Descartes to his own day. 
Therein Schleiermacher, for instance, gets more space than the 
Stoics, and Descartes as much as all Scholasticism. R. Adamson 
(2.21) allots no less that sixteen pages to Kant, but only five to 
the whole period between the death of Aristotle and Bacon, com- 
prising the Megarians, Stoics, Commentators and Scholastics. A few 
years ago Max Polenz gave barely a dozen pages to Stoic logic in his 
big book on this school (22.2). 

Along with this basic attitude went a misunderstanding of ancient 
logical teaching. It was consistently treated as though exhibiting 
nothing what corresponded with the content of 'classical' logic; all 
else either went quite unnoticed or was interpreted in the sense of 
the 'classical' syllogistic, or again, written off as mere subtlety. It is 
impossible to discuss the details of these misinterpretations, but at 
least they should be illustrated by some examples. 

The Aristotelian assertoric syllogistic is distortedly present in 
'classical' i.e. Ockhamist style (34.01) as a rule of inference with the 
immortal 'Socrates' brought into the minor premiss, whereas for 
Aristotle the syllogism is a conditional propositional form (§13) 
without any singular terms. Stoic logic was throughout absurdly 
treated as a term logic (2.23), whereas it was quite plainly a pro- 
positional logic (§20). Aristotelian modal logic (§15) was so little 
understood that when A. Becker gave the correct interpretation of its 
teaching in 1934 (2.24), his view was generally thought to be revo- 
lutionary, though in essence this interpretation is quite elementary 
and was known to Albert the Great (33.03). Aristotle and Thomas 
Aquinas were both credited with the Theophrastan analysis of 
modal propositions and modal syllogisms, which they never advo- 
cated (2.25). 

No wonder then that with the rise of mathematical logic theorems 
belonging to the elementary wealth of past epochs were saddled with 
the names of De Morgan, Peirce and others; there was as yet no 
scientific history of formal logic. 

C. RESEARCH IN THE 20th CENTURY 

Scientific history of formal logic, free from the prejudices we have 
mentioned and based on a thorough study of texts, first developed in 
the 20th century. The most important researches in the various 
fields are referred to in the relevant parts of our survey. Here we 
shall only notice the following points. 

The rise of modern history of logic concerning all periods save 
the mathematical was made possible by the work of historians 
of philosophy and philologists in the 19th century. These published 
for the first time a series of correct texts edited with reference to 
their context in the history of literature. But the majority of ancient 

9 



A HISTORY OF FORMAL LOGIC 

philologists, medievalists and Sanskrit scholars had only slight 
understanding of and little interest in formal logic. History of 
logic could not be established on the sole basis of their great and 
laborious work. 

For its appearance we have to thank the fact that formal logic 
took on a new lease of life and was re-born as mathematical. Nearly 
all the more recent researches in this history were carried out by 
mathematical logicians or by historians trained in mathematical 
logic. We mention only three here: Charles Sanders Peirce, the fore- 
runner of modern research, versed in ancient and scholastic logic; 
Heinrich Scholz and Jan Lukasiewicz, with their publications of 
1931 and 1935 (2.26, 2.27), both exercising a decisive influence 
on many parts of the history of logic, thanks to whom there have 
appeared serious studies of ancient, medieval and Indian logic. 

But still we have only made a start. Though we are already in 
possession of basic insights into the nature of the different historical 
varieties of formal logic, our knowledge is still mostly fragmentary. 
This is markedly the case for Scholastic and Indian logic. But as 
the history of logic is now being systematically attended to by a 
small group of researchers it can be foreseen that this state of 
affairs will be improved in the coming decades. 



§3.THE EVOLUTION OF FORMAL LOGIC 

As an introduction to the present state of research and to justify 
the arrangement of this book, a summary presentation of results is 
now needed. The view we present is a new one of the growth of 
formal logic, stated here for the first time. It is a view which markedly 
diverges not only from all previous conceptions of the history of 
logic, but also from opinions that are still widespread about the 
general history of thought. But it is no 'synthetic a priori judgment', 
rather is it a position adopted in accordance with empirical findings 
and based on the total results of the present book. Its significance 
seems not to be confined within the boundaries of the history of 
logic: the view might be taken as a contribution to the general 
history of human thought and hence to the sociology of knowledge. 

A. CONCERNING THE GEOGRAPHY AND CHRONOLOGY OF LOGIC 

Formal logic, so far as we know, originated in two and only two 
cultural regions: in the west and in India. Elsewhere, e.g. in China, 
we do occasionally find a method of discussion and a sophistic 
(3.01), but no formal logic in the sense of Aristotle or Dignaga was 
developed there. 

Both these logics later spread far beyond the frontiers of their 
native region. We are not now speaking merely of the extension of 

10 



EVOLUTION OF FORMAL LOGIC 

European logic to America, Australia and other countries settled 
from Europe; North America, for instance, which from the time 
of Peirce has been one of the most important centres of logical 
research, can be treated as belonging to the western cultural region. 
Rather it is a matter of western logic having conquered the Arabian 
world in the high Middle Age, and penetrated Armenian culture 
through missionaries.* Other examples could be adduced. The 
same holds for Indian logic, which penetrated to Tibet, China, 
Japan and elsewhere. Geographically, then, we arc concerned with 
two vital centres of evolution for logic, whose influence eventually 
spread far abroad. 

On the subject of the chronology of logic and its division into 
periods there is this to be said : this history begins in Europe in the 
4th century B.C., in India about the 1st century a.d. Previously 
there is in Greece, India and China, perhaps also in other places, 
something like a pre-history of logic; but it is a complete mistake 
to speak of a 'logic of the Upanishads' or a 'logic of the Pythagoreans'. 
Thinkers of these schools did indeed establish chains of inference, 
but logic consists in studying inference, not in inferring. No such 
study can be detected with certainty before Plato and the Nyaga; 
at best we find some customary, fixed and canonical rules of dis- 
cussion, but any complete critical appreciation and analysis of 
these rules are missing. 

The history of western logic can be divided into five periods: 
1. the ancient period (to the 6th century a.d.); 2. the high Middle 
Age (7th to 11th centuries); 3. the Scholastic period (11th to loth 
centuries); 4. the older period of modern 'classical' logic (16th to 
19th centuries); 5. mathematical logic (from the middle of the 19th 
century). Two of those are not creative periods — the high Middle 
Age and the time of 'classical' logic, so that they can be left almost 
unnoticed in a history of problems. The hypothesis that there was 
no creative logical investigation between the ancient and Scholastic 
periods might very probably be destroyed by a knowledge of Arabian 
logic, but so far little work has been done on this, and as the results 
of what research has been undertaken are only to be found in Arabic, 
they are unfortunately not available to us. 

Indian logic cannot so far be divided into periods with comparable 
exactness. It only seems safe to say that we must accept at least 
two great periods, the older Nyaya and Buddhism up to the 10th 
century of our era, and the Navya (new) Nyaya from the 12th 
century onwards. 



* I am grateful to Prof. M. van den Oudenrijn for having drawn my attention 
to this fact. 



11 



A HISTORY OF FORMAL LOGIC 
B. HOW LOGIC EVOLVED 

Logic shows no linear continuity of evolution. Its history resembles 
rather a broken line. From modest beginnings it usually raises 
itself to a notable height very quickly — within about a century — 
but then the decline follows as fast. Former gains are forgotten, the 
problems are no longer found interesting, or the very possibility 
of carrying on the study is destroyed by political and cultural 
events. Then, after centuries, the search begins anew. Nothing 
of the old wealth remains but a few fragments; building on those, 
logic rises again. 

We might therefore suppose that the evolution of logic could be 
presented as a sine-curve; a long decline following on short periods 
of elevation. But such a picture would not be exact. The 'new' logic 
which follows on a period of barbarism is not a simple expansion of 
the old ; it has for the most part different presuppositions and points 
of view, uses a different technique and evolves aspects of the 
problematic that previously received little notice. It takes on a 
different shape from the logic of the past. 

That holds in the temporal dimension for western and, with some 
limitations, for Indian logic. Perhaps it also holds in the spatial 
dimension for the relation between the two considered as wholes. 
We can indeed aptly compare Indian logic with ancient and Scholas- 
tic logic in Europe, as lacking the notion of calculation; but beyond 
that there is hardly any resemblance. They are different varieties 
of logic. It is difficult to fit the Indian achievements into a scheme 
of evolution in the west. 

The essential feature of the whole history of logic seems then to be 
the appearance of different varieties of this science separated both 
in time and space. 

C. THE VARIETIES OF LOGIC 

There are in essence, so far as we can determine, four such forms: 

1. The Ancient Variety of Logic. In this period logical theorems 
are mostly formulated in the object-language, and semantics is in 
being, though undeveloped. The logical formulae consist of words 
of ordinary language with addition of variables. But this ordinary 
language is as it were simplified, in that the chief words in it occur 
only in their immediate semantic function. The basis of this logic 
is the thought as expressed in natural language, and the syntactical 
laws of the language are presupposed. It is from this material that 
the ancient logicians abstract their formal laws and rules. 

2. The Scholastic Variety of Logic. The Scholastics began by linking 
themselves to antiquity, and thus far simply took over and developed 
what was old. But from the end of the 12th century they started to 
construct something entirely new. This logic which is properly 

12 



EVOLUTION OF FORMAL LOGIC 

their own is almost all formulated metalogically. It is based on and 
accompanied by an accurate and well-developed semantics. For- 
mulae consist of words from ordinary language, with very few or no 
variables, but there results no narrowing of the semantic functions 
as in antiquity. Scholastic logic is accordingly a thorough-going 
attempt to grasp formal laws expressed in natural language (Latin 
with plentifully differentiated syntactical rules and semantic 
functions. As in ancient logic, so here too we have to do with 
abstraction from ordinary language. 

3. The Mathematical Variety of Logic. Here we find a certain 
regress to the ancient variety. Till a fairly late date (about 1930; 
mathematical logic is formulated purely in the object-language, 
with rich use of variables; the words and signs used have narrowly 
limited semantic functions; semantics remains almost unnoticed 
and plays not nearly so marked a role as in the Middle Ages and 
after its resurgence since about 1930. Mathematical logic introduces 
two novelties; first, the use of an artifical language; second, and 
more important, the constructive development of logic. This last 
means that the system is first developed formalistically and only 
afterwards interpreted, at least in principle. 

Common to the three western varieties of logic is a far-reaching 
formalism and preponderantly extensional treatment of logical laws. 

4. The Indian Variety of Logic. This differs from the western in 
both the characteristics just mentioned. Indian logic succeeds is 
stating certain formal laws, but formalism is little developed and is 
obviously considered to be subsidiary. Again the standpoint is 
preponderantly intensional in so far as the Indian logicians of the 
last period knew how to formulate a highly developed logic of 
terms without employing quantifiers. 

The fore-going arrangement is schematic and oversimplified, 
especially in regard to ancient and Indian logic. One could ask, for 
instance, whether Megarian-Stoic logic really belongs to the same 
variety as Aristotelian, or whether it is on the contrary mainly 
new, having regard to its markedly semantic attitude. 

Still more justified would perhaps be the division of Indian logic 
into different forms. One could find, for example, considerable 
justification for saying that Buddhist logic differs notably from the 
strict Nyaya tradition not only in its philosophical basis, nor only 
in details, but with this big difference that the Buddhists show a 
manifestly extensional tendency in contrast to the Nyaya com- 
mentators. Again, evidence is not lacking that the Navy a Nyaya 
does not properly exhibit a quite new type of logic, since in some 
doctrines, as in the matter of Vyapti, it takes over Buddhist modes 
of expression, in others it follows the Nyaya tradition, in others 
again it develops a new set of problems and takes up a fresh stand- 
point. 

13 



A HISTORY OF FORMAL LOGIC 

However the difference between Aristotle and the Megarian- 
Stoic school seems hardly significant enough to justify speaking of 
two different forms of logic. As to Indian logic our knowledge is so 
incomplete that it would be rash to draft a division and charac- 
terization of its different forms. 

A further problem that belongs here is that of the so-called 
'classical' logic. One could understand it as a distinct variety, since 
while it consists of fragments of scholastic logic (taking over for 
example the mnemonic Barbara, Celarenl etc., yet these fragments 
are interpreted quite unscholastically, in an ancient rather than 
scholastic way. But the content of this logic is so poor, it is loaded 
with so many utter misunderstandings, and its creative power is so 
extremely weak, that one can hardly risk calling something so 
decadent a distinct variety of logic and so setting it on a level with 
ancient, scholastic, mathematical and Indian logic. 



D. THE UNITY OF LOGIC 

We said above that every new variety of logic contains new 
logical problems. It is easy to find examples of that: in Scholasticism 
there are the magnificent semiotic investigations about the proprie- 
taies terminorum, then the analysis of propositions containing time- 
variables, investigations about quantifiers, etc; in mathematical 
logic the problems of multiple quantification, description, logical 
paradoxes, and so on. It is evident that quite different systems of 
formal logic are developed as a result. To be sure, that also sometimes 
happens within the framework of a single form of logic, as when we 
single out Theophrastan modal logic as different from Aristotle's. 
The class of alternative systems of formal logic has increased 
greatly especially since Principia Mathematica. 

One might therefore get the impression that the history of logic 
evidences a relativism in logical doctrine, i.e. that we see the rise 
of different logics. But we have spoken not of different logics, rather 
of different varieties of one logic. This way of speaking has been 
chosen for speculative reasons, viz. that the existence of many 
systems of logic provides no proof that logic is relative. There is, 
further, an empirical basis for speaking of one logic. For history 
shows us not only the emergence of new problems and laws but 
also, and perhaps much more strikingly, the persistent recurrence 
of the same set of logical problems. 

The following examples may serve to support this thesis: 

1. The problem of implication. Posed by the Megarians and Stoics 
(20.05 ff.), it was resumed by the Scholastics (30.09 ff.), and again by 
the mathematical logicians (41.11 ff.). Closely connected with it, 
so it seems, was what the Indians called vyapli (53.20, 54.07 f.). 
Perhaps more remarkable is the fact that the same results were 

14 



EVOLUTION OF FORMAL LOGIC 

reached quite independently in different periods. Thus material 
implication is defined in just the same way by Philo (20.07), Burleigh 
(30.14) and Peirce (41.12 f.), in each case by means of truth-values. 
Another definition is also first found among the Megarians (20.10j, 
again, and this time as their main concept of implication, among the 
Scholastics (30.11 f.), and is re-introduced by Lewis in 1918 (49.04). 

2. The semantic paradoxes serve as a second example. Already 
posed in the time of Aristotle (23.18), discussed by the Stoics 
(23.20), the problem of these is found again in the Scholastics 
(35.05 ff.), and forms one of the main themes in mathematical 
logic (§ 48). Re-discovery of the same solutions is again in evidence 
here, e.g. Russell's vicious-circle principle was already known to 
Paul of Venice. 

3. A third group of problems common to western logic is that of 
questions about modal logic. Posed by Aristotle (§ 15), these ques- 
tions were thoroughly gone into by the Scholastics (§ 33) and have 
taken on a new lease of life in the latest phase of mathematical 
logic (49.03). 

4. We may refer again to the analysis of quantifiers : the results of 
Albert of Saxony and Peirce are based on the same understanding 
of the problem and run exactly parallel. 

5. Similar correspondences can be noticed between Indian and 
western logic. D. H. H. Ingalls has recently discovered a long 
series of problems and solutions common to the two regions. Most 
remarkable is the fact that Indian logic, evolving in quite different 
conditions from western, and independently of it, eventually dis- 
covered precisely the scholastic syllogism, and, as did western logic, 
made its central problem the question of 'necessary connection'. 

Still further examples could be adduced in this connection; it 
seems as though there is in the history of logic a set of basic problems, 
taken up again and again in spite of all differences of standpoint, and, 
still more important, similarly solved again and again. 

Itis not too easy to express exactly, but every reader who is a 
logician will see unmistakeably the community of mind, by which 
we mean the recurrent interest in certain matters, the way and 
style of treating them, among all inquirers in the field of what we 
comprehend within the different forms of formal logic. Read in 
conjunction our texts 16.19, 22.16fL, 31.22, 33.20, 41.11 ff. There 
can be no doubt that the same attitude and spirit is expressed in 
them all. 

E. THE PROBLEM OF PROGRESS 

Closely connected with the question of the unity of logic is the 
difficult problem of its progress. One thing is certain: that this 
problem cannot be solved a priori by blind belief in the continuous 
growth to perfection of human knowledge, but only on the basis of 

15 



A HISTORY OF FORMAL LOGIC 

a thoroughly empirical inquiry into detail. We can only learn 
whether logic has progressed in the course of its history from that 
history itself. We cannot discover it by means of a philosophic 
dogma. 

But the problem is not easily solvable with our present historical 
knowledge. One question which it involves seems indeed to be safely 
answerable, but the requisites for dealing with others are still 
lacking. 

We can safely state the following: 

1. The history of logic shows, as has already been remarked, no 
linear ascending development. Consequently in the case of an advance, 
it can only take place firstly, within a given period and form of 
logic, and secondly, so as to raise the later forms to a higher level 
than the earlier. 

2. Some advance within single periods and forms of logic is 
readily perceivable. We can see it best in Indian, but also in Scho- 
lastic and mathematical logic. Every particular of these periods 
affords a safe criterion of progress; each of them has its essential 
problems, and by comparing their formulation and solution in 
different logicians of the same period we can easily see that the 
later writers pose the questions more sharply, apply better method 
to their solution, know more laws and rules. 

3. If the history of logic is considered as a whole, here too a certain 
advance can be established with safety. This consists in the fact 
that new problems are forthcoming in the later forms of logic. 
Thus for example the highly wrought semiotic problematic of the 
Scholastics is quite new in comparison with that of antiquity, and 
therefore also more complete; the logical paradoxes (not the 
semantic ones) of the mathematical logicians are new; so too Albert 
of Saxony's problem of defining quantifiers is new. These are again 
only some examples from the many possible ones. 

On the other hand, the following question seems to be still 
undecidable in the present state of knowledge: taking logic as a 
whole, is every later form superior to all earlier ones ? 

Too often this question is answered affirmatively with an eye 
on mathematical logic, particularly because people compare it with 
its immediate predecessor, 'classical' logic, and are struck by the 
mass of laws and rules which calculation makes available in the 
new form. 

But 'classical' logic is by no means to be equated with the whole 
of older logic; it is rather a decadent form of our science, a 'dead 
period' in its evolution. Calculation, again, is certainly a useful 
tool for logic, but only as facilitating new insights into logical 
interconnection. It is undeniable that such insights, e.g. in the logic 
of relations, have been reached by its means, and the convenience 
and accuracy of this instrument are so great that no serious logician 

16 



EVOLUTION OF FORMAL LOGIC 

can now dispense with it. But we would not go so far as to say that 
calculation has at every point allowed mathematical logic to surpass 
the older forms. Think for example of two-valued prepositional 
logic: the essentially new features introduced by Principia Mathe- 
matical are quite unimportant when we compare the scholastic 
treatment. 

Once again the matter reduces to our insufficient knowledge of 
the earlier forms of logic. For years people spoke of a supposed 
great discovery by De Morgan; then Lukasiewicz showed that his 
famous law was part of the elementary doctrine of Scholasticism. 
The discovery of truth-matrices was ascribed to Peirce, or even 
Wittgenstein; Peirce himself found it in the Megarians. D. Ingalls 
found Frege's classical definition of number in the Indian Mathu- 
ranatha (17th century). And then we are all too well aware that we 
know, as has been said, only fragments of Scholastic and Indian 
logic, while much more awaits us in manuscripts and even in unread 
printed works. The Megarian-Stoic logic, too, is lost, except for a 
few poor fragments transmitted by its opponents. 

Also highly relevant to the question of the continual progress of 
logic throughout its history is the fact that the earlier varieties 
are not simply predecessors of contemporary logic, but deal in part 
with the same or similar problems though from a different stand- 
point and by different methods. Now it is hard for a logician trained 
in the contemporary variety of logic to think himself into another. 
In other words, it is hard for him to find a criterion of comparison. 
He is constantly tempted to consider valuable only what fits into 
the categories of his own logic. Impressed by our technique, which is 
not by itself properly logic, having only superficial knowledge of 
past forms, judging from a particular standpoint, we too often risk 
misunderstanding and under-rating other forms. 

Even in the present state of knowledge we can be sure that 
various points about the older forms still escape our comprehension. 
One example is the Scholastic doctrine of supposition, which is 
evidently richer in important insights and rules than the semiotic so 
far developed by mathematical logic. Another is perhaps the 
treatment of implication (vyapti) by the thinkers of the Navya 
Nyaya. Still further examples could be given. 

Again, when an unprejudiced logician reads some late-Scholastic 
texts, or it may be some Stoic fragments, he cannot resist the im- 
pression that their general logical level, their freedom of movement 
in a very abstract domain, their exactness of formulation, while they 
are equalled in our time, have by no means been excelled. The modern 
mathematical logician certainly has a strong support in his calculus, 
but all too frequently that same calculus leads him to dispense 
with thought just where it may be most required. A conspicuous 
example of this danger is provided by statements made for long 

17 



A HISTORY OF FORMAL LOGIC 

years by mathematical logicians concerning the problem of the null 
class. 

These considerations tell against the thesis that logic has pro- 
gressed as a whole, i.e. from variety to variety; it looks as though we 
have insufficient grounds for holding it. But of course it does not at 
all follow that another thesis, viz. of a purely cyclic development of 
formal logic with continual recurrence of the same culminating 
points, is sufficiently established. 

The historian can only say; we do not know whether there is an 
over-all progress in the history of logic. 



§4. METHOD AND PLAN 
A. HISTORY OF PROBLEMS, AND DOCUMENTATION 

Conformably to the directions of the series Orbis Academicus this 
work will present a documented history of problems. 

We are not, therefore, presenting a material history of logic 
dealing with everything that has any historical importance, but a 
delineation of the history of the problematic together with the 
complex of essential ideas and methods that are closely connected 
with it. We only take into account those periods which have made an 
essential contribution to the problematic, and among logicians those 
who seem to us to rank as specially good representatives of their 
period. In this connection some thinkers of outstanding importance, 
Aristotle above all, Frege too, will receive much fuller treatment than 
would be permissible in a material history. 

The story will be told with the help of texts, and those originally 
written in a foreign language have been translated into English. 
This procedure, unusual in a scientific work, is justified by the consi- 
deration that only a few readers could understand all the texts if 
they were adduced in their original language. For even those readers 
with some competence in Greek are not automatically able to under- 
stand with ease a text of formal logic in that tongue. But the spe- 
cialist logician will easiliy be able to find the original text by reference 
to the sources. 

The passages quoted will be fairly thoroughly commented where 
this seems useful, for without some commentary many of them 
would not be readily intelligible. 

B. PLAN OF THE WORK 

In itself such a history admits of being arranged according to 
problems. One could consider first questions of semiotics, then 
propositional ones, then those of predicate logic etc., so as to pursue 
the whole history of each class of problems. E.g. the chapter on 
propositional logic could begin with Aristotle, go on to the Megarian- 

18 



METHOD AND PLAN 

Stoic theory of Xoyo<;, then to the scholastic consequentiae, to the 
propositional interpretation of the Boolean calculus, to McColl, Peirce 

and Frege, to chapters 2 — 5 of the Principia, finally to Lukasiewicz. 

Such a method of treatment is, however, forbidden by the non- 
linear evolution of logic, and above all by the fact that it takes on a 
different form in every epoch. For every particular group of pro- 
blems within one variety is closely connected with other complexes 
of problems in the same variety. Torn from its context and ranged 
with the cognate problems in another variety, it would be, not just 
unintelligible, but quite misunderstood. The problem of impli- 
cation provides a good example: the Scholastics put it in the 
context of their theory of meaning, and their theory is not to In- 
understood apart from that. Every problem considered in a given 
variety of logic needs viewing in the context of the total problematic 
of that variety. 

It is necessary, therefore, to arrange the whole history according 
to the varieties of logic. Within each we have tried to show the 
connection of the various groups of problems. This has, however, not 
proved to be the best course everywhere. In the discussion of 
antiquity a grouping of the material according to the chronology of 
logicians and schools has seemed preferable, especially because one 
logician, Aristotle, has an incomparably great importance. 

C. CHARACTER OF THE CONTENTS 

As our knowledge of many domains is still very fragmentary, we 
cannot aim at completeness. One period that is probably fairly 
important, the Arabian, cannot be noticed at all. Citations from 
Scholasticism are certainly only fragments. Even our knowledge of 
ancient and mathematical logic is far from satisfactory. Conse- 
quently this work serves rather as a survey of some aspects of the 
history of logical problems than as a compilation of all that is essen- 
tial to it. 

What is rather aimed at is a general orientation in whatever kind 
of problems, methods and notions is proper to each variety of logic, 
and by that means some presentation of the general course of the 
history of logic and its laws. The emphasis will be put on this course 
of the problematic as a whole. 

Hence we have also decided to risk a short account of Indian 
logic, in spite of subjective and objective reasons to the contrary. 
For this logic seems to be of great interest precisely with reference to 
the laws of the whole development. At the same time it is the only 
form which has developed quite independently of the others. The 
chapter on Indian logic must, however, be managed differently 
from the rest, not only because our knowledge of the subject is even 
less sufficient than of Scholastic logic, but also because we have to 
rely on translations. This chapter will be treated as a kind of appendix. 

19 



A HISTORY OF FORMAL LOGIC 

§5. TERMINOLOGY 

In order to establish a comparison between the problems and 
theorems which have been formulated in different epochs and 
languages, we have had to use a unified terminology in our comments. 
For the most part we have taken this from the vocabulary of 
contemporary formal logic. But as this vocabulary is not at all 
familiar to the majority of readers, we shall here explain the most 
important technical expressions. 

A. TECHNICAL EXPRESSIONS 

By 'expression', 'formula', 'word', 'symbol' etc. we here intend 
what Morris calls the sign-vehicle, and so the material component 
of the sign ; i.e. a certain quantity of ink, or bundle of sound waves. A 
specially important class of expression is that of sentences, i.e. 
expressions which can be characterized as true or false. It must be 
stressed that a sentence, so understood, is an expression, a material 
sign, and not what that signs stands for. The word 'proposition' has 
been variously used, as synonymous with 'sentence' in the sense 
just explained (cf. 26.03), more normally for a sentence precisely as 
meaningful (Scholastics generally), sometimes with various psycho- 
logical and subjective connotations (cf. the 'judgment' of the 
'classical' logicians), nowadays commonly as the objective content of 
a meaningful sentence (cf. the Stoic a^icopia). In our commentaries 
we keep 'sentence' for the material expression, as above and use 
'proposition' in the sense appropriate to the historical context and as 
indicated by normal usage, which seems frequently to approximate 
to that of the Scholastics. 

We divide expressions into atomic and molecular (the thought is 
Aristotelian, cf. 10.14 and 10.24), the former being without parts 
that are themselves expressions of the given language, the latter con- 
taining such parts. Molecular expressions are analysed sometimes 
into subject and predicate in accordance with the tradition of Ari- 
stotle and the Scholastics, sometimes into functor and argument. 
The functor is the determining element, the argument the one deter- 
mined ; this is also true of predicate and subject respectively, but the 
other pair of terms is more general in applicability. 'And', 'not', 
names of relations, are thought of as functors. 

We distinguish between constant and variable expressions (again 
with Aristotle, cf. 13.04), called constants and variables for short. The 
former have a determinate sense, the latter only serve to mark void 
places in which constants can be substituted. Thus, for example, in 
'x smokes', V is a variable and 'smokes' a constant. With Frege 
(42.02) we call a molecular expression which exhibits a variable a 
function. Thus we speak of propositional functions, that is to say of 

20 



TERMINOLOGY 

expressions which, if the variables that occur in them are properly 
replaced by constants, become sentences (or propositions in the 
Scholastic sense). l x smokes' is such a propositional function. 

Among propositional functions we often mention the logical 
sum or inclusive disjunction of two propositions or terms, the logical 
product or conjunction, implication and equivalence. Quantifiers (cf. 
44.01), 'all', 'some', 'for every x\ 'there is a y such that', are some- 
times counted as functors. 

Variables which can only be meaningfully replaced by propositions 
we call propositional variables; such as can only be meaningfully 
replaced by terms we call term-variables. Correspondingly we speak 
of laws of propositional logic and term-logic. Term-logic is divided 
into predicate-, class- (or classial), and relation-logic. Predicate- 
logic treats of intensions, class-logic of extensions; relation-logic is 
the theory of those special formal properties which belong to rela- 
tions, e.g. symmetry (if Ft holds between a and 6, then it also holds 
between b and a), transitivity (if B holds between a and b and be- 
tween b and c, then it also holds between a and c) etc. 

The general doctrine of signs we call, with W. Morris (5.01), 
semioiic. This is divided into syntax (theory of the relationships 
between signs), semantics (theory of the relationships between signs 
and their significates), and pragmatics (theory of the relationships 
between signs and those who use them). Correspondingly we speak 
of syntactical, semantic and pragmatic laws and theories. In the 
field of semantics we distinguish between the denotation and the 
meaning or sense of a sign — which denotes the object of which it 
is a sign (its reference), and means its content. (In translating 
Scholastic texts we use 'signifies' for 'significaV and leave further 
determination to be judged, where possible, by the context.) Thus 
for example, the word 'horse' denotes a horse, but means what 
makes a horse a horse, what we might call 'horseness'. We dis- 
tinguish further between object-language in which the signs denote 
objects that are not part of the language, and the corresponding 
meta-language in which the signs denote those of the object-language. 
In accordance with this terminology the word 'cat' in the sentence 
'a cat is an animal' belongs to the object-language since it denotes 
a non-linguistic object, but in the sentence '"cat" is a substantive' 
it belongs to the meta-language, since it denotes the word 'cat' and 
not a cat itself. When an expression is used as the name of another 
expression that has the same form, we follow the prescription of 
Frege (39.03) and write it between quotation-marks. 

Finally we distinguish between logical laws and rules, as did the 
Stoics (§ 22, A and B) and Scholastics (cf. the commentary on 31.13). 
Laws state what is the case, rules authorize one to proceed in such 
and such a way. 



21 



A HISTORY OF FORMAL LOGIC 

B. CONCERNING MATHEMATICO-LOGICAL SYMBOLISM 

In divergence from the widespread practice, which is that of the 
author himself, all use of mathematico-logical symbolism has been 
avoided in the commentaries on texts not of this character. In many 
cases this symbolism affords easy abbreviation, and laws formulated 
by its means are much easier for the specialist to read than verbally 
expressed propositional functions or propositions. But two reasons 
militate against its use: 

1. First, objectively, it introduces an appreciable risk of misunder- 
standing the text. Such a risk is present in every case of translation, 
but it is particularly great when one uses a terminology with so 
narrowly defined a sense as that of mathematical logic. Take, for 
example, signs of implication. Those at our disposal essentially 
reduce to two: 'd' and 'F'. Which of them are we to use to express 
Diodorean implication? Certainly not the first, for that means 
Philonian implication; but not the second either, for that would 
mean that one was sure that Diodorus defined implication just like 
Lewis or Buridan, which is by no means certain. Another example 
is the Peano-Russellian paraphrase of Aristotelian syllogistic as it 
occurs in the Principia (5.02). It is undoubtedly a misinterpretation 
of Aristotle's thought, for it falsifies many laws of the syllogistic 
which on another interpretation (that of Lukasiewicz) can be seen 
to be correct (5.03). 

Some notions not deriving from mathematical logic could indeed 
be expressed in its symbolism, e.g. the Philonian implication or that 
of Buridan; but to single out these for such interpretation and to 
make use of verbal formulation in other cases would be to cause a 
complication that is better avoided. 

But of course that is not to say that no such symbolism ought 
to be employed for any form of logic. For particular logicians, or 
a particular form, the use of an artificial symbolism is not only 
possible, but to be desired. But then every case requires a special 
symbolism. What we cannot do is to create a unique symbolism 
suitable for all the ideas that have been developed in the different 
varieties of logic. 

2. A subjective reason is provided by the limits of the work, 
which aims to make allowance for the reader who is formed in the 
humanities but innocent of mathematics. For such, and they are 
obviously the majority, mathematico-logical symbolism would 
not clarify his reading, but cloud it unnecessarily. 

In these circumstances we have been at pains to use such texts 
as exhibit no artificial symbolism, even in the chapter on mathe- 
matical logic, so far as that is possible. Symbolic texts are of course 
cited as well, and in such a way that one who wishes to acquire the 
symbolic language of mathematical logic can learn the essentials 
from this work. But the texts which treat of the basic problems of 

22 



TERMINOLOGY 



logic have been chosen in such a way that they are as far as possible 
intelligible without a knowledge of this symbolism. 



C. TYPOGRAPHICAL CONVENTIONS 

All texts are numbered decimally, the integral part giving the 
paragraph in which the citation occurs, the decimals referring to a 
consecutive numbering within the paragraph. 

Texts are set in larger type than the commentaries, except for 
formulas due to the author, which are also in larger type. 

Added words are enclosed in round parentheses. Expressions in 
square parentheses occur thus in the text itself. Formulas are an 
exception to this: all parentheses occurring in them, together with 
their contents, occur so in the original texts. 

Quotation marks and italics in ancient and scholastic texts are 
due to the author. 

Remarks concerning textual criticism are presented in starred 
footnotes. 

Special points concerning the chapter on Indian logic are stated 
in § 50, D. 



23 



PART II 

The Greek Variety of Logic 



§6. INTRODUCTION TO GREEK LOGIC 

A. LOGICIANS IN CHRONOLOGICAL ORDER 

Aristotle, the first historian of philosophy, calls Zeno of Elea the 
'founder of dialectic' (6.01), but the first two men, so far as we know, 
to reflect seriously on logical problems were Plato and Euclid of 
Megara, both pupils of Socrates. And as Aristotle himself ascribes 
to Socrates important services in the domain of logic (6.02), or 
rather of methodology from which logic later developed, perhaps 
Socrates should be considered to be the father of Greek logic. 

Aristotle was a pupil of Plato, and his logic undoubtedly grew out 
of the practice of the Platonic Academy. Aristotle's chief pupil 
and long-time collaborator, Theophrastus, provides the link between 
the logical thought of his master and that of the Stoa. For con- 
temporaneously and parallel with Aristotelian logic there developed 
that derived from Euclid, of which the first important representatives 
were Megarians, Diodorus Cronus, Philo of Megara and others; 
later came the Stoics, who were closely connected with the Megarians, 
having Chrysippus as their most important thinker. 

After the death of Chrysippus, disputes arose between the Peripa- 
tetic and Megarian-Stoic schools, the latter now represented by the 
Stoics alone, and syncretism became prominent. Even then logicians 
were not lacking, the more important among them being apparently 
the commentators on Aristotle's logical works (Alexander, Philo- 
ponus), many Sceptics (especially Sextus Empiricus), these in the 
3rd century B.C., and finally Boethius (5th-6th century a.d.). 

The following table shows the chronological and doctrinal 
connection down to Chrysippus: 

Zeno of Elea, c. 464/60 b.c. 



Socrates ob. 399 The ancient Sophists 



Plato 428/7-348/47 Euclid of Megara c. 400 



4- 
Aristotle 384-322 

Theophrastus ob. 287/86 



4- 
■Diodorus Cronus ob. 307 



rJL/lUUUl US Kj 
Philo o 



f Megara 



* Zeno of Citium 

336/35-246/43 

4- 

Chrysippus of Soli 

281/78-208/05 
Peripatetic School Stoic School Megarian School 



Syncretism 



26 



INTRODUCTION TO GREEK LOGIC 
B. PERIODS 

The problematic of formal logic by and large began with Aristotle. 
He was undoubtedly the most fertile logician there has ever been, 
in the sense that a great many logical problems were raised for the 
first time in his works. Close to him in the history of ancient logic 
is a group of thinkers who are nearly as important, the Megarian- 
Stoic school. Aristotle lived in the 4th century B.C.; the essential 
development of the Megarian-Stoic school can be thought of as 
ending with the death of Ghrysippus of Soli at the end of the 3rd 
century B.C. Hence in Greek antiquity there is a relatively short 
period to be considered, from the second half of the 4th to the end 
of the 3rd century B.C. 

But that does not mean that there was no logical problematic 
outside those 150 years. Even before Aristotle, a problematic 
emerged in the form of the pre-Socratic and Platonic dialectic, 
admittedly without ever developing into a logical theory. Again, 
long after the death of Chrysippus, and right on to the end of anti- 
quity, i.e. to the death of Boethius (6th century a.d.), many reflec- 
tions on logical problems are to be found in the so-called Commen- 
tators. This last period is not comparable in fruitfulness with that of 
Aristotle and the Stoics, but we are indebted to it for various 
insights worth remark. 

Accordingly, from our point of view, antiquity is divided into 
three main periods: 

1. the preparatory period, to the time when Aristotle began to 
edit his Topics. 

2. the Aristotelian-Megarian-Stoic period, occupying the second 
half of the 4th to the end of the 3rd century B.C. 

3. the period of the Commentators, from about 200 b.c. to the 
death of Boethius at the beginning of the 6th. century a.d. 

The second of these periods is so outstandingly important that 
it is appropriate to divide it into two sections covering respectively 
Aristotle and the Megarian-Stoic school. We have then four tempo- 
rally distinct sections: 1. pre-Aristotelians, 2. Aristotle and his 
immediate pupils, 3. the Megarian-Stoic school, 4. Commentators. 



C. STATE OF RESEARCH 

The history of Greek logic is the relatively best-known period in 
the development of formal logic. By contrast with the Middle 
Ages and after, and to some extent with logistic too, nearly all the 
surviving texts of the logicians of this age are readily available in 
good modern editions, together with a whole series of scientific 
treatises on their contents. In this connection there are two classes 
of works: 



27 



THE GREEK VARIETY OF LOGIC 

a) On the one hand the philologists have been busy for more 
than a century with solving numerous and often difficult problems 
of literary history relevant to ancient logic. Yet great as is the debt 
of gratitude owed by logicians to this immense work, one cannot 
pass over the fact that most philologists lack training in formal logic 
and so too often overlook just the most interesting of the ancient 
texts. Mostly, too, their interest centres on ontological, metaphysical, 
epistemological and psychological questions, so that logic comes to 
be almost always neglected. To quote only one example: logic is 
allotted few pages in Polenz's two big volumes on the Stoa. Then 
again editions made without a thorough logical training are often 
insufficient: Kochalsky's edition of Stoic fragments may serve as 
an instance. 

b) On the other hand logicians too, especially since the pioneer 
work of G. Vailati (1904) and A. Rustow (1908), have considered a 
fair number of problems arising from these texts. Epoch-making 
in this field is the article Zur Geschichle der Aussagenlogik (1935) 
by J. Lukasiewicz. The same scholar has given us books on the 
principle of contradiction in Aristotle and the Aristotelian (assertoric) 
syllogistic. Important too are the researches of H. Scholz whose 
Geschichle der Logik appeared in 1931 and who has written a number 
of other studies. Each of these has formed a small school. J. Sala- 
mucha investigated the concept of deduction in Aristotle (1930). 
I. M. Bochenski wrote a monograph on Theophrastus (1939); his 
pupils J. Stakelum (1940) and R. van den Driessche (1948) published 
studies on the period of the Commentators, the former dealing with 
Galen, the latter with Boethius. Boethius has also been dealt with 
by K. Diirr (1952). A. Becker, a pupil of Scholz, produced an impor- 
tant work on Aristotle's modal syllogisms (1933). B. Mates, influen- 
ced by Lukasiewicz, has made a thorough study of Stoic logic (1953). 

The state of inquiry up to now may be characterized thus: 
Aristotelian studies are well opened up, though much is still missing, 
e.g. discussion of the Topics; good editions of the text are also 
available. We also have a very fair knowledge of Megarian-Stoic 
logic, though fresh editions of the texts are desirable. Very little 
work has been done on the period of the Commentators, but good 
editions are mostly to hand. The pre-Aristotelian period is also very 
insufficiently explored, notwithstanding the valuable studies by 
A. Krokiewicz, a philologist with logical training. Especially 
desirable is a thorough-going treatment of the beginnings of logic 
in Plato, though admittedly such a work would meet with con- 
siderable difficulties. 

More exact information about the literature will be found in the 
Bibliography. 



28 



I. THE PRECURSORS 



7. THE BEGINNINGS 



When Aristotle brought to a close the earliest part of his logical 

work, i.e. the Topics and De Sophislicis Elenchis, he could proudly 
write : 

7.01 In the case of all discoveries the results of previous 
labours that have been handed down from others have been 
advanced bit by bit by those who have taken them on, 
whereas the original discoveries generally make an advance 
that is small at first though much more useful than the develop- 
ment which later springs out of them. For it may be that in 
everything, as the saying is, 'the first start is the main part' : 
and for this reason also it is the most difficult; . . . 

Of this inquiry, on the other hand, it was not the case that 
part of the work had been thoroughly done before, while part 
had not. Nothing existed at all. 

A. TEXTS 

What Aristotle says of 'this inquiry' of his seems still to hold good ; 
we know of no logic, i.e. an elaborated doctrine of rules or laws, 
earlier than the Topics. Certain rules of inference, however, appear 
to have been consciously applied long before Aristotle by many 
Greeks, without being reflectively formulated, much less axiomatized. 
Aristotle himself says elsewhere that Zeno of Elea was the 'founder 
of dialectic' (6.01), and it is in fact hardly possible that Zeno 
formulated his famous paradoxes without being aware of the rules 
he was applying. The texts ascribed to him are only to be found in 
late commentators, including, however, Simplicius who was a 
serious investigator; criticism casts no doubt on their authenticity. 
We give some examples of his dialectic : 

7.02 In the case that they (beings) are many, they must 
be as many as they are, neither more nor less. But if they are 
as many as they are, then they are limited (determinate). If 
(however) beings are many, then they are unlimited (indeter- 
minate) : since there are yet other beings between the beings 
and others again between those. And thus beings are unlimited 
(indeterminate). 

7.03 If beings are, every one must have magnitude and volume, 
and one part of it must be distinct from another . . . And so, 
if they are many they must be at once small and great; small, 

29 



THE GREEK VARIETY OF LOGIC 

since they have no magnitude, and great since they are un- 
limited (indeterminate). 

7.04 If there is a place, it is in something; for every being 
is in something; but what is in something is also in a place. 
Hence the place will itself be in a place, and so on without 
end; hence there is no place. 

G. Vailati stressed a text from Plato in which a similar process of 
inference is used : 

7.05 Socrates: And the best of the joke is, that he acknow- 
ledges the truth of their opinion who believe his own opinion 
to be false; for he admits that the opinions of all men are true. 

Theodorus: Certainly. 

Socrates: And does he not allow that his own opinion is 
false, if he admits that the opinion of those who think him 
false is true ? 

Theodorus : Of course. 

Socrates: Whereas the other side do not admit that they 
speak falsely? 

Theodorus: They do not. 

Socrates: And he, as may be inferred from his writings, 
agrees that this opinion is also true. 

Theodorus: Clearly. 

Socrates: Then all mankind, beginning with Protagoras, will 
contend, or rather, I should say that he will allow, when he 
concedes that his adversary has a true opinion, Protagoras, 
I say, will himself allow that neither a dog nor any ordinary 
man is the measure of anything which he has not learned - am 
I not right? 

Theodorus: Yes. 

The big fragment of Gorgias (7.06) also contains something 
similar, but this is so evidently composed in the technical terminology 
of the Stoics and betrays so highly developed a technique of logical 
thought that we cannot ascribe it to the Sophists, nor even to 
Aristotle. It is, however, possible that the young Aristotle did indeed 
formulate the famous proof of the necessity of philosophy in the way 
which it ascribed to him. This proof is transmitted to us in the follow- 
ing three passages among others: 

7.07 There are cases in which, whatever view we adopt, we 
can refute on that ground a proposition under consideration. 
So for instance, if someone was to say that it is needless to 
philosophize : since the enquiry whether one needs to philo- 

30 



PRECURSORS 

sophize or not involves philosophizing, as he (Aristotle) has 
himself said in the Protreplicus, and since the exercise of 
a philosophical pursuit is itself to philosophize. In showing 
that both positions characterize the man in every case, we 
shall refute the thesis propounded. In this case one can rest 
one's proof on both views. 

7.08 Or as Aristotle says in the work entitled Protreplicus 
in which he encourages the young to philosophize. For he 
says: if one must philosophize, then one must philosophize; if 
one does not have to philosophize, one must still philosophize. 
So in any case one has to philosophize. 

7.09 Of the same kind is the Aristotelian dictum in the 
Protreplicus: whether one has to philosophize or not, one 
must philosophize. But either one must philosophize or not; 
hence one must in any case philosophize. 



B. SIGNIFICANCE 

All the texts adduced above spring from the milieu of 'dialectic'. 
This word that is later given so many meanings and is so mis-used 
originally had the same meaning as our 'discussion'. It is a matter 
of disputation between two speakers or writers. That is probably the 
reason why most of the rules of inference used here - termed, as it 
seems, iogoi' - lead to negative conclusions: the purpose was to 
refute something, to show that the assertion propounded by the 
opponent is false. 

This suggests the conjecture that these logoi belong to the field 
of propositional logic, that is to say that it is here a matter of 
logical relations between propositions as wholes without any ana- 
lysis of their structure. And in fact the pre-Aristotelian logoi were 
often so understood. However, this interpretation seems untenable: 
Aristotle himself was aware of the very abstract laws of propositional 
logic only exceptionally and at the end of his scientific career; so 
much the less ought we to ascribe this - Megarian-Stoic - manner of 
thought to the pre-Aristotelians. We have rather to do with certain 
specifications of general rules of propositional logic. Thus these 
dialecticians were not thinking of, for example, the abstract scheme 
of propositional logic corresponding to modus ponendo ponens: 

7.101 If p, then q; but p ; therefore q: 
but rather of the more special law 

7.102 If A belongs to x, then B also belongs to x; but .4. 
belongs to x; therefore B also belongs to x. 

31 



THE GREEK VARIETY OF LOGIC 

We purposely omit quantifiers here, since while such were necessa- 
rily present to the thought obscurely, at this level there can be no 
question of a conscious acceptance of such logical apparatus. 

We note further, that at the level of pre-Aristotelian dialectic, it is 
always a matter of rules not of laws; they are principles stating how 
one should proceed, not laws, which describe an objective state of 
affairs. That does not mean of course that the dialecticians were in 
any way conscious of the distinction between the two; but from our 
point of view, what they used were rules. 

This said, we can interpret as follows the several logoi previously 
adduced. For each we give the logical sentence corresponding to the 
rule of inference which it employs. 

Zeno quoted by Simplicius (7.02, 03, 04) : 

7.021 If A belongs to x then B and C also belong to x; but 
B and C do not belong to x; therefore neither does A belong 
to x. 

7.022 Suppose that if A belongs to x, B also belongs to x 
and if B belongs to x, C also belongs to x, then if A belongs to 
x, C also belongs to x. 

Plato in the Theaeletus (7.05) : 

7.051 If A belongs to x then A does not belong to x; 
therefore A does not belong to x. 

Closer examination of that last item shows that it is much more 
complex and belongs to the realm of metalogic. Plato's thought 
proceeds after this fashion: the proposition propounded by Protago- 
ras means: for every x, if x says 'p', then p. Let us abbreviate that 
by *S\ Now there is some (at least one) x who says that S is not the 
case. Therefore S is not the case. Therefore if S, then not S. From 
which it follows in accordance with 7.051, that S is not the case. 
While Plato certainly did not expressly draw this conclusion, he 
evidently intended it. 

Aristotle quoted by Alexander (7.07) : 

7.071 Suppose that if A belongs to x, A belongs to x, and 
if A does not belong to x, A belongs to a?, then A belongs to x. 

The anonymous scholiast has a fuller formula (7.09) : 

7.091 If A belongs to aj, then A belongs to x; if A does not 
belong to x, then A belongs to x; either A belongs to x or A 
does not belong to x; therefore A belongs to x, 
but whether it actually occurred in Aristotle may be doubted. 
Possibly the Protreplicus contained merely the simple formula, 
transmitted by Lactantius : 

32 



PRECURSORS 

7.092 If A does not belong to x, then A belongs to x; 
therefore A belongs to x. 

A series of similar formulae underlie the processes to be found in 
the great Gorgias-fragment (7.06), but these appear to be so markedly 
interpreted in the light of Stoic logic that we have no guarantee of 
anything genuinely due to the sophist himself. 



§ 8. PLATO 

While Plato, in respect of many rules used in his dialectic, 
belongs to the same period as Zeno (as too does the youthful Ari- 
stotle), he begins something essentially new in our field, and that 
from several points of view. 

A. CONCEPT OF LOGIC 

In the first place Plato rendered the immortal service of being the 
first to grasp and formulate a clear idea of logic. The relevant text 
occurs in the Timaeus and runs : 

8.01 God invented and gave us sight to the end that we 
might behold the courses of intelligence in the heaven, and 
apply them to the courses of our own intelligence which are 
akin to them, the unperturbed to the perturbed; and that we, 
learning them and partaking of the natural truth of reason, 
might imitate the absolutely unerring courses of God and 
regulate our own vagaries. 

Such a conception of logic was, however, only possible for Plato, 
because he was, as it seems, the originator of another quite original 
idea, namely that of universally necessary laws (granting that he 
depended in this on the logos-doctrine of Heracleitus and other 
earlier thinkers). The concept of such laws is closely connected with 
Plato's theory of ideas, which itself developed through reflection on 
Geometry as it then existed. The whole post-Platonic western 
tradition is so penetrated with these ideas, that it is not easy for a 
westerner to grasp their enormous significance. Evidently no formal 
logic was possible without the notion of universally valid law. From 
this point of view the importance of Plato for the history of logic 
can best be seen when we consider the development of the science in 
India, i.e. in a culture which had to create logic without a Plato. One 
can see in the history of Indian logic that it took hundreds of years 
to accomplish what was done in Greece in a generation thanks to 
the elan of Plato's genius, namely to rise to the standpoint of uni- 
versal validity. 

33 



THE GREEK VARIETY OF LOGIC 

We cannot here expound Plato's doctrine of ideas, as it belongs to 
ontology and metaphysics, and is further beset with difficult pro- 
blems of literary history. 

B. APPROACHES TO LOGICAL FORMULAE 

Plato tried throughout his life to realize the ideal of a logic as laid 
down above, but without success. The tollowing extracts from his 
dialectic, in which he makes a laboured approach to quite simple 
laws, show how difficult he found it to solve logical questions that 
seem elementary to us. 

8.02 Socrates: Then I shall proceed to add, that if the 
temperate soul is the good soul, the soul which is in the 
opposite condition, that is, the foolish and intemperate, is 
the bad soul. - Very true. - And will not the temperate man 
do what is proper, both in relation to the gods and to men; - 
for he would not be temperate if he did not? - Certainly he 
will do what is proper. 

8.03 Socrates: Tell me, then, - Is not that which is pious 
necessarily just? 

Euthyphro: Yes. 

Socrates: And is, then, all which is just pious? or, is that 
which is pious all just, but that which is just, only in part and 
not all, pious? 

Euthyphro: I do not understand you, Socrates. 

8.04 When you asked me, I certainly did say that the 
courageous are the confident; but I was never asked whether 
the confident are the courageous; if you had asked me, I 
should have answered 'Not all of them :' and what I did answer 
you have not proved to be false, although you proceeded to 
show that those who have knowledge are more courageous 
than they were before they had knowledge, and more coura- 
geous than others who have no knowledge, and were then led 
on to think that courage is the same as wisdom. But in this 
way of arguing you might come to imagine that strength is 
wisdom. You might begin by asking whether the strong are 
able, and I should say 'Yes'; and then whether those who 
know how to wrestle are not more able to wrestle than those 
who do not know how to wrestle, and more able after than 
before they had learned, and I should assent. And when I had 
admitted this, you might use my admissions in such a way as 
to prove that upon my view wisdom is strength; whereas 
in that case I should not have admitted, any more than in the 

34 



PRECURSORS 

other, that the able are strong, although I have admitted that 

the strong are able. For there is a difference between ability 
and strength; the former is given by knowledge as well as by 
madness or rage, but strength comes from nature and a 
healthy state of the body. And in like manner J say of confi- 
dence and courage, that they are not the same; and I argue 
that the courageous are confident, but not all the confident 
courageous. For confidence may be given to men by art, and 
also, like ability, by madness and rage; but courage comes to 
them from nature and the healthy state of the soul. 

In the first of these texts is involved the (false) thesis: Suppose, if 
A belongs to x, B also belongs to x, then: if A does not belong to x, 
then B does not belong to x. The second shows the difficulties found 
concerning the convertibility of universal affirmative sentences: viz. 
whether 'all B is A' follows from 'all A is B ' . The third text shows 
still more clearly how hard Plato felt these questions to be; it further 
has the great interest that, to show the invalidity of the fore- 
going rule of conversion, he betakes himself to complicated extra- 
logical discussions - about bodily strength, for instance. 

C. DIAERESIS 

Yet Plato's approximations were not without fruit. He seems to 
have been the first to progress from a negative dialectic to the con- 
cept of positive proof; for him the aim of dialectic is not to refute 
the opinions of opponents but positive 'definition of the essence'. In 
this he definitely directed attention to the logic of predicates, which 
is probably the cause of Aristotelian logic taking the form it did. 
The chief goal which Plato set himself was to discover essences, i.e. to 
find statements which between them define what an object is. For 
this he found a special method - the first logical, consciously ela- 
borated inferential procedure known to us - namely his famous 
'hunt' for the definition by division (Sioctpeais). How thoroughly 
conscious he was of not only using such a method but of endeavour- 
ing to give it the clearest possible formulation, we see in the cele- 
brated text of the Sophist in which the method, before being practis- 
ed, is applied in an easy example: 

8.05 Stranger: Meanwhile you and I will begin together and 
enquire into the nature of the Sophist, first of the three : I 
should like you to make out what he is and bring him to light 
in a discussion; for at present we are only agreed about the 
name, but of the thing to which we both apply the name 
possibly you have one notion and I another; whereas we ought 
always to come to an understanding about the thing itself in 

35 



THE GREEK VARIETY OF LOGIC 

terms of a definition, and not merely about the name minus the 
definition. Now the tribe of Sophists which we are investi- 
gation is not easily caught or defined ; and the world has long 
ago agreed, that if great subjects are to be adequately treated, 
they must be studied in the lesser and easier instances of them 
before we proceed to the greatest of all. And as I know that the 
tribe of Sophists is troublesome and hard to be caught, I 
should recommend that we practise beforehand the method 
which is to be applied to him on some simple and smaller thing, 
unless you can suggest a better way. 

Theaetetus: Indeed I cannot. 

Stranger: Then suppose that we work out some lesser 
example which will be a pattern of the greater? 

Theaetetus : Good. 

Stranger: What is there which is well known and not great, 
and is yet as susceptible of definition as any larger thing? Shall 
I say an angler? He is familiar to all of us, and not a very 
interesting or important person. 

Theaetetus : He is not. 

Stranger: Yet I suspect that he will furnish us with the sort 
of definition and line of enquiry which we want. 

Theaetetus: Very good. 

Stranger: Let us begin by asking whether he is a man having 
art or not having art, but some other power. 

Theaetetus: He is clearly a man of art. 

Stranger: And of arts there are two kinds? 

Stranger: Seeing, then, that all arts are either acquisitive or 
creative, in which class shall we place the art of the angler? 

Theaetetus : Clearly in the acquisitive class. 

Stranger: And the acquisitive may be subdivided into two 
parts: there is exchange, which is voluntary and is effected by 
gifts, hire, purchase; and the other part of acquisitive, which 
takes by force of word or deed, may be termed conquest? 

Theaetetus: That is implied in what has been said. 

Stranger: And may not conquest be again subdivided? 

Theaetetus : How? 

Stranger: Open force may be called fighting, and secret 
force may have the general name of hunting? 

Theaetetus : Yes. 

Stranger: And there is no reason why the art of hunting 
should not be further divided. 

36 



PRECURSORS 

Theaetetus : How would you make the division? 

Stranger: Into the hunting of living and of lifeless prey. 

Theaetetus :Yes, if both kinds exist. 

Stranger: Of course they exist; but the hunting after life- 
less things having no special name, except some sorts of 
diving, and other small matters, may be omitted; the hunting 
after living things may be called animal hunting. 

Theaetetus : Yes. 

Stranger: And animal hunting may be truly said to have two 
divisions, land-animal hunting, which has many kinds and 
names, and water-animal hunting, or the hunting after ani- 
mals who swim? 

Theaetetus: True. 

Stranger: And of swimming animals, one class lives on the 
wing and the other in the water? 

Theaetetus : Certainly. 

Stranger: Fowling is the general term under which the 
hunting of all birds is included. 

Theaetetus : True. 

Stranger: The hunting of animals who live in the water has 
the general name of fishing. 

Theaetetus : Yes. 

Stranger: And this sort of hunting may be further divided 
also into two principal kinds? 

Theaetetus : What are they? 

Stranger: There is one kind which takes them in nets, another 
which takes them by a blow. 

Theaetetus: What do you mean, and how do you distinguish 
them ? 

Stranger : As to the first kind - all that surrounds and encloses 
anything to prevent egress, may be rightly called an enclosure. 

Theaetetus: Very true. 

Stranger: For which reason twig baskets, casting-nets, 
nooses, creels, and the like may all be termed 'enclosures'? 

Theaetetus: True. 

Stranger: And therefore this first kind of capture may be 
called by us capture with enclosures, or something of that sort ? 

Theaetetus: Yes. 

Stranger: The other kind, which is practised by a blow with 
hooks and three-pronged spears, when summed up under one 
name, may be called striking, unless you, Theaetetus, can 
find some better name ? 

37 



THE GREEK VARIETY OF LOGIC 

Theaetetus : Never mind the name - what you suggest will 
do very well. 

Stranger: There is one mode of striking, which is done at 
night, and by the light of a fire, and is by the hunters them- 
selves called firing, or spearing by firelight. 

Theaetetus: True. 

Stranger: And the fishing by day is called by the general 
name of barbing, because the spears, too, are barbed at the 
point. 

Theaetetus: Yes, that is the term. 

Stranger: Of this barb-fishing, that which strikes the fish 
who is below from above is called spearing, because this is 
the way in which the three-pronged spears are mostly used. 

Theaetetus : Yes, it is often called so. 

Stranger: Then now there is only one kind remaining. 

Theaetetus : What is that ? 

Stranger: When a hook is used, and the fish is not struck in 
any chance part of his body, as he is with the spear, but only 
about the head and mouth, and is then drawn out from below 
upwards with reeds and rods: - What is the right name of 
that mode of fishing, Theaetetus? 

Theaetetus : I suspect that we have now discovered the object 
of our search. 

Stranger: Then now you and I have come to an under- 
standing not only about the name of the angler's art, but 
about the definition of the thing itself. One half of all art was 
acquisitive - half of the acquisitive art was conquest or 
taking by force, half of this was hunting, and half of hunting 
was hunting animals, half of this was hunting water animals - 
of this again, the under half was fishing, half of fishing was 
striking; a part of striking was fishing with a barb, and one 
half of this again, being the kind which strikes with a hook 
and draws the fish from below upwards, is the art which we 
have been seeking, and which from the nature of the opera- 
tion is denoted angling or drawing up. 

Theaetetus: The result has been quite satisfactorily brought 
out. 

The process is evidently not conclusive: as Aristotle has forcibly 
shown (8.06), it involves a succession of assertions, not a proof; it 
may be helpful as a method, but it is not formal logic. 

Formal logic is reserved for Aristotle. But a close examination of 
the contents of his logical works assures us that everything contained 

38 



PRECURSORS 

in the Organon is conditioned in one way or another by the practice 
of Platonism. The Topics is probably only a conscious elaboration of 
the numerous logoi current in the Academy; even the Analytics, 
invention of Aristotle's own as it was, is evidently based on 
'division', which it improved and raised to the level of a genuine 
logical process. That is the second great service which Plato 
rendered to formal logic: his thought made possible the emergence 
of the science with Aristotle. 



39 



II. ARISTOTLE 

§9. THE WORK OF ARISTOTLE AND THE PRO- 
BLEMS OF ITS LITERARY HISTORY 

The surviving logical works of Aristotle set many difficult pro- 
blems of literary history which as yet are only partly solved. They are 
of outstanding importance for the history of the problems of logic, 
since within the short span of Aristotle's life formal logic seems to 
have made more progress than in any other epoch. It is no exagge- 
ration to say that Aristotle has a unique place in the history of 
logic in that 1. he was the first formal logician, 2. he developed formal 
logic in at least two (perhaps three) different forms, 3. he consciously 
elaborated some parts of it in a remarkably complete way. Further- 
more, he exercised a decisive influence on the history of logic for 
more than two thousand years, and even today much of the doctrine 
is traceable back to him. It follows that an adequate understanding 
of the development of his logical thought is of extreme importance 
for an appreciation of the history of logical problems in general, and 
particularly of course for western logic. 

A. WORKS 

The surviving works of the Stagirite were set in order and edited 
by Andronicus of Rhodes in the first century b.c. The resulting 
Corpus Aristotelicum contains, as to logical works, first and fore- 
most what was later called the Organon, comprising: 

1. The Categories, 

2. About Propositions (properly: About Interpretation; we shall 
use the title Hermeneia), 

3. The Prior Analytics, two books: A and B, 

4. The Posterior Analytics, two books : A and B, 

5. The Topics, eight books: A,B, T, A, E, Z, H, 0, 

6. The Sophistic Refutations, one book. 

Besides these, the whole fourth book (T) of the so-called Meta- 
physics is concerned with logical problems, while other works, e.g. the 
Rhetoric and Poetics contain occasional points of logic. 

B. PROBLEMS 

The most important problems concerning the Organon are the 
following : 

1. Authenticity 

In the past the genuineness of all Aristotle's logical writings has 
often been doubted. Today, apart from isolated passages and 
perhaps individual chapters, the Categories alone is seriously con- 
sidered to be spurious. The doubt about the genuineness of the 

40 



A R I S T O T L E 

Hermeneia seems not convincing. The remaining works rank by and 
large as genuine. * 

2. Character 

Should we view the logical works of Aristotle as methodically 
constructed and systematic treatises? Researches made hitherto 
allow us to suppose this only for some parts of the Organon. The 
Hermeneia and Topics enjoy the relatively greatest unity. The 
Prior Analytics are evidently composed of several strata, while the 
Posterior Analytics are mainly rather a collection of notes for lectures 
than a systematic work. But even in those parts of the Organon that 
are systematically constructed later additions are to be found here 
and there. 

3. Chronology 

The Organon, arranged as we have it, is constructed on a syste- 
matic principle: the Categories treats of terms, the Hermeneia of 
propositions, the remaining works of inference: thus the Prior 
Analytics treats of syllogisms in general, the three other works 
successively of apodeictic (scientific), dialectical and sophistical 
syllogisms. For this systematization Andronicus found support in 
the very text of the Organon; e.g. at the beginning of the Prior 
Analytics it is said (9.02) that the syllogism consists of propositions 
(npOTCcaic,) , these of terms (6po«;). In the Topics (9.03) and also in 
the Prior Analytics (9.04) the syllogism is analysed in just that way. 
At the end of the Sophistic Refutations occurs the sentence already 
cited (7.01), which appears to indicate that this work is the latest 
of Aristotle's logical works. 

It is also not impossible that at the end of his life Aristotle himself 
drafted an arrangement of his logic and accordingly ordered his 
notes and treatises somewhat as follows. But of course this late 
systematization has, to our present knowledge, little to do with the 
actual development of this logic. 

We have no extrinsic criteria to help us establish the chronolo- 
gical sequence of the different parts of the Organon. On the other 
hand their content affords some assistance, as will now be briefly 
explained.** 



* The thesis of Josef Ziircher (9.01) that nearly all the formal logic in the 
Organon is due not to Aristotle but to his pupil Theophrastus, is not worth 
serious consideration. 

* * Chr. Brandis opened up the great matter of the literary problems of the 
Organon in his paper Vber die Reihenfolge des aristotelischen Organons (9.05); the 
well-known work of W. Jager (9.06) contributed important insights; its basic 
pre-suppositions were applied to the Organon by F. Solmsen (9.07). Solmsen's 
opinions were submitted to a thorough criticism by Sir W. D. Ross (9.08) with 
an adverse result in some cases. Important contributions to the chronology of 
the Organon are to be found in A. Becker (9.09) and J. Lukasiewicz ^9.10). 

41 



THE GREEK VARIETY OF LOGIC 

a. Chronological criteria 

aa) A first criterion to determine the relative date of origin is 
afforded by the fact that the syllogism in the sense of the Prior 
Analytics (we shall call it the 'analytical syllogism') is completely 
absent from several parts of the Organon. But it is one of the most 
important discoveries, and it can hardly be imagined that Aristotle 
would have failed to make use of it, once he had made it. We 
conclude that works in which there are no analytic syllogisms are 
earlier than those in which they occur. 

bb) In some parts of the Organon we find variables (viz. the letters 
A, B, r, etc.), in others not. But variables are another epoch- 
making discovery in the domain of logic, and the degree to which it 
impressed Aristotle can be seen in the places where he uses and 
abuses them to the point of tediousness. Now there are some works 
where variables would be very useful but where they do not occur. 
We suppose that these works are earlier than those where they do 
occur. 

cc) The third criterion - afforded by the technical level of the 
thought - cannot, unlike the first two, be formulated simply, but is 
apparent to every experienced logician at his first perusal of a text. 
From this point of view there are big differences between the 
various passages of the Organon: in some we find ourselves at a still 
very primitive level, reminiscent of pre-Socratic logic, while in 
others Aristotle shows himself to be the master of a strictly formal 
and very pure logical technique. One aspect of this progress appears 
in the constantly developing analysis of statements: at first this 
is accomplished by means of the simple subject-predicate schema 
(S — P), then quantifiers occur ( l P belongs [does not belong] to all, 
to none, to some S'), finally we meet a subtle formula that reminds 
us of the modern formal implication : 'All that belongs to S, belongs 
also to P\ This criterion can be formulated as follows: the higher 
and more formal the technique of analysis and proof, so much the 
later is the work. 

dd) Modal logic corresponds much better with Aristotle's own 
philosophy (which contains the doctrine of act and potency as an 
essential feature) than does purely assertoric logic in which the 
distinction between act and potency obtains no expression. Asser- 
toric logic fits much better with the Platonism to which Aristotle 
subscribed in his youth. Accordingly we may view those writings 
and chapters containing modal logic as having been composed later. 

ee) Some of these criteria can be further sharpened. Thus we can 
trace some development in the theory of analytic syllogism. Again, 
Aristotle seems to have used letters at first as mere abbreviations 
for words and only later as genuine variables. Finally one can detect 
a not insignificant progress in the structure of modal logic. 

It may certainly be doubted whether any one of these criteria is 

42 



ARISTOTLE 

of value by itself for establishing the chronology. But when all, or 
at least several of them point in the same direction, the resulting 
sequence seems to enjoy as high a degree of probability as is ever 
possible in the historical sciences. 

b. Chronological list 

The application of these criteria enable us to draw up the following 

chronological list of Aristotle's logical writings: 

aa) The Topics (together with the Categories if this is to be accep- 
ted as genuine) undoubtedly comes at the start. There is to be found 
in it no trace of the analytic syllogism, no variables, no modal logic, 
and the technical level of the thought is relatively low. While the 
Sophistic Refutations simply forms the last book of the Topics, it 
appears to have been composed a little later. Book F of the Meta- 
physics probably belongs to the same period. The Topics and the 
Refutations together contain Aristotle's first logic. The remark at 
the end of the Sophistic Refutations about the 'whole' of logic 
refers to that elaboration. 

bb) The Hermeneia and - perhaps - book B of the Posterior 
Analytics form a kind of transitional stage: the syllogistic can be 
seen emerging. In the Hermeneia we hear nothing of syllogism and 
there are no variables. Both, but evidently only in an early stage 
of development, occur in Posterior Analytics book B. The technical 
level of thought is much higher than in the Topics. The Hermeneia 
also contains a doctrine of modality, which is, however, quite 
primitive compared with that in the Prior Analytics. 

cc) Book A of the Prior Analytics, with the exception of chapters 
8-22, contains Aristotle's second logic, a fully developed assertoric 
syllogistic. He is by now in possession of a clear idea of analytic 
syllogism, uses variables with sureness, and moves freely at a rela- 
tively high technical level. The analysis of propositions has been 
deepened. Missing, as yet, are modal logic, and reflective considera- 
tion of the syllogistic system. Perhaps book A of the Posterior 
Analytics may be ascribed to the same period. Solmsen made this 
the first of all the analytic books, but W. D. Ross's arguments 
against this seem convincing. (The latter holds that book B of the 
Posterior Analytics is also later than the Prior.) 

dd) Finally we may ascribe to a still later period chapters 8-22 of 
book A, which contains the modal syllogistic logic, and book B of 
the Prior Analytics. These can be said to contain Aristotle's third 
logic, which differs less from the second than does the second from 
the first. We find here a developed modal logic, marred admittedly 
by many incompletenesses and evidently not finished, and also 
penetrating remarks, partly metalogical, about the system of 

43 



THE GREEK VARIETY OF LOGIC 

syllogistic. In them Aristotle offers us insights into formal logic of 
remarkable subtlety and acuteness. He states too some theorems 
of propositional logic with the aid of propositional variables. 

Of course there can be no question of absolute certainty in answer- 
ing the chronological questions, especially as the text is corrupt in 
many places or sprinkled with bits from other periods. It is only 
certain that the Topics and Sophistic Refutations contain a different 
and earlier logic than the Analytics, and that the Hermeneia exhi- 
bits an intermediate stage. For the rest we have well-founded 
hypotheses which can lay claim at least to great probability. 

In accordance with these hypotheses we shall speak of three 
logics of Aristotle. 



§10. CONCEPT OF LOGIC. SEMIOTIC 

A. NAME AND PLACE OF LOGIC 

Aristotle has no special technical name for logic: what we now 
call 'logical' he calls 'analytic' (ocvocXutlxo^: 10.01) or 'following 
from the premisses' (ex tgW xeiuivcov: 10.02), while the expression 
'logical' (Xoyix6<;) means the same as our 'probable' (10.03) or 
again 'epistemological'. 

10.04 Of propositions and problems there are - to com- 
prehend the matter in outline - three divisions : for some are 
ethical propositions, some are on natural philosophy, while 
some are logical. . . such as this are logical, e.g. 'Is the know- 
ledge of opposites the same or not?' 

The question whether logic is a part of philosophy or its instrument 
(opyavov) - and hence an art - is nowhere raised by Aristotle in the 
extant works. 

B. THE SUBJECT-MATTER OF LOGIC 

Yet Aristotle knew well enough what he demanded of logic. 
That appears from the model statements of the subject-matter of 
his logical treatises. For instance he says in the Topics: 

10.05 First then we must say what reasoning is, and what 
its varieties are, in order to grasp dialectical reasoning: for 
this is the object of our search in the treatise before us. Now 
reasoning is an argument in which, certain things being laid 
down, something other than these necessarily comes about 
through them. It is a 'demonstration' when the premisses 
from which the reasoning starts are true and primary . . . 

44 



ARISTOTLE 

reasoning, on the other hand, is 'dialectical', if it reasons 
from opinions that are generally accepted. . . . Again, reason- 
ing is 'contentious' if it starts from opinions that seem to be 
generally accepted, but are not really such, or again if it 
merely seems to reason from opinions that are or seem to be 
generally accepted. 

Compare the following text from the Prior Analytics: 

10.06 After these distinctions we now state by what means, 
when, and how every syllogism is produced; subsequently 
we must speak of demonstration. Syllogism should be discuss- 
ed before demonstration, because syllogism is the more 
general: demonstration is a sort of syllogism, but not every 
syllogism is a demonstration. 

The thought is perfectly clear: Aristotle is looking for relations 
of dependence which authorize necessary inference, and in that 
connection makes a sharp distinction between the validity of this 
relation and the kind of premisses, or their truth. The text contains 
what is historically the first formulation of the concept of a formal 
logic, universally valid and independent of subject-matter. 

Accordingly it is syllogism which is the subject of logic. This is 
a form of speech (Xoyo^) consisting of premisses (nporuikaeu;) 
themselves composed of terms (5poi). 'Premiss' and 'term' are thus 
defined by Aristotle: 

10.07 A premiss is a form of speech which affirms or 
denies something of something. ... A term I call that into 
which the premiss is resolved, that is to say what is predicated 
and that of which it is predicated by means of the addition 
of being or not being. 

What emerges from that text is the complete neutrality of the 
technical expressions 'term', 'premiss', 'syllogism', relative to any 
philosophical interpretation. For the premiss consists of terms, the 
syllogism of premisses, and premisses are logoi, which can equally 
well mean utterances or thoughts or objective contents, so that the 
way is open to a formalist, psychological or objectivist interpreta- 
tion. All these interpretations are permissible in regard to Aristo- 
telian logic; the purely logical system excludes none of them. Guided 
by his original intuition the founder of formal logic so chose his 
terminology as to rise above the clash of interpretations to the level 
of pure logic. 

However if one considers Aristotelian logic in its entirety, it is 
easy to see that this neutrality is not the result of a lack of interest in 
problems of interpretation, but is on the contrary an abstraction 

45 



THE GREEK VARIETY OF LOGIC 

from a complex semiotic doctrine. In some places Aristotle seems to 
plead for a psychological type of theory, as when, for example, he 

says: 

10.08 All syllogism and therefore a fortiori demonstration, 
is addressed not to outward speech but to that within the 
soul. 

At the same time it must be said that he attaches great importance 
to the 'outward speech', since he elaborates a well-developed 
theory of logical syntax and many points of semantic interest. 
All this teaching, which is next to be considered, warrants the con- 
clusion that the practice of Aristotelian logic was undoubtedly to 
regard meaningful words as its subject-matter. 



C. SYNTAX 

Aristotle is the founder of logical syntax, following here some 
hints of the Sophists and Plato. He sketched the first attempt known 
to us at a system of syntactical categories. For we find in the Her- 
meneia an explicit division of the parts of speech into atomic (nouns 
and verbs) and molecular (sentences). 

10.09 By a noun we mean a sound significant by con- 
vention, which has no reference to time, and of which no 
part is significant apart from the rest. 

10.10 A verb is that which, in addition to its proper 
meaning, carries with it the notion of time. No part of it has 
any independent meaning, and it is a sign of something said 
of something else. 

This theory is further supplemented by discussions of cases and 
inflexions of words, and by considerations about negated nouns and 
verbs. 

10.11 A sentence is a significant portion of speech, some 
parts of which have an independent meaning, that is to say, 
as an utterance (^olgiq) , though not as the expression of any 
positive judgement (xaTowpaais). 

10.12 The first unified declarative sentence is the affirma- 
tion; the next, the denial. All other sentences are unified 
by combination. 

Anticipating the further explanations, we may summarize the 
whole scheme of syntactical categories as presented in the Herme- 
neia, after this fashion : 

46 



A FU S T O T L E 



conven- 
tionally 
significant 
sound 
(10.13) 



atomic 

(10.14) 



mole- 
cular 

Xoyoc, 

(10.24) 



noun 
(6vo[i.a) in 
broad 
sense 



other (poLGZic, 



statement 
qltzo^olvgic, 
(10.25) 



noun in 

strict 

sense (10.10) 



simple 

composite 
(10.17) 

individual 



miversal 
10.18] 

negated noun (10.19) 
cases of noun (10.20) 
verb p7)[i.a (10.21) 
negated verb (10.22) 
inflexions of verbs (10.23) 

affirmative 

xaxacpac!.? 

negative 

OLTCOyOLGIC, 



atomic 



(10.26) 

with singu- 
lar subject 

with 

universal 

subject 



taken not taken 

universally universallv 

(10.27) 
molecular (10.26) 



other Xoyoi, (also called cpfxaeic,) (10.26) 



This schema underlies the whole development of logical syntax, 
and semantics too, until the rise of mathematical logic. Only this 
last will introduce anything essentially new: the attempt to treat 
syntactical categories by means of an artificial language. Aristotle 
on the other hand, and with him the Stoic and scholastic traditions, 
sought to grasp the syntactical structure of ordinary language. 

D. SEMANTICS 

The texts previously cited already contain points belonging to 
the domain of semantics. The general principle is thus formulated by 
Aristotle: 



47 



THE GREEK VARIETY OF LOGIC 

10.28 Spoken words are the symbols of mental experience 
and written words are the symbols of spoken words. 

It follows that thoughts are themselves symbols of things. 
Aristotle lays great stress on the parallelism between things, 
thoughts and symbols, and correspondingly develops two important 
semiotic theories: 

10.29 Things are said to be named 'equivocally' when, 
though they have a common name, the definitions correspond- 
ing with the name differs for each. Thus a real man and a 
figure in a picture can both lay claim to the name 'animal'. . . . 

Things are said to be named 'univocally' which have both 
the name and the definition answering to the name in com- 
mon. A man and an ox are both 'animal' .... 

Things are said to be named 'derivatively', which derive 
their name from some other name, but differ from it in 
termination. Thus the grammarian derives his name from the 
word 'grammar' and the courageous man from the word 
'courage'. 

Equivocity must be excluded from demonstrations, since it 
leads to fallacies (10.30). Elsewhere Aristotle distinguishes various 
kinds of equivocity. 

10.31 The good, therefore, is not some common element 
answering to one Idea. But what then do we mean by the 
good ? It is surely not like the things that only chance to have 
the same name. Are goods one, then, by being derived from 
one good or by all contributing to one good, or are they 
rather one by analogy? Certainly as sight is in the body, so 
is reason in the soul, and so on in other cases. 



This division can be presented schematically as follows: 



equivocal 
expressions 



in strict sense 
dcTco tux"/)*; 
(accidentally equivocal 

in broad sense 
(systematically equivocal) 



from one (deep* hoc,) 
to one (npbq lv) 

by proportion 

(xoct' avaXoytav) 



48 



ARISTOTLE 

In the Metaphysics and Hermeneia we find a clear sern ioric 
theory of truth : 

10.32 For falsity and truth are not in things - it is not as if 
the good were true and the bad were in itself false - but in 
thought. 

10.33 As there are in the mind thoughts which do not 
involve truth or falsity, and also those which must be either 
true or false, so it is in speech. For truth and falsity imply com- 
bination and separation. Nouns and verbs, provided nothing 
is added, are like thoughts without combination or separation. 

10.34 Yet not every sentence states something, but only 
those in which there is truth or falsity, and not all are of 
that kind. Thus a prayer is a sentence, but is neither true 
nor false . . . the present theory is concerned with such sen- 
tences as are statements (#7co<pocvTix&s X6yo<;). 



Aristotle also constructs a definition of truth by means of equi- 
valences: 

10.35 If it is true to say that a thing is white, it must 
necessarily be white; if the reverse proposition is true, it 
will of necessity not be white. Again, if it is white, the proposi- 
tion stating that it is white was true; if it is not white, the 
proposition to the opposite effect was true. And if it is not 
white , the man who states that it is is making a false statement ; 
and if the man who states that it is white is making a false 
statement, it follows that it is not white. 



11. THE TOPICS 



A. SUBJECT AND PURPOSE 

The Topics contains Aristotle's first logic, and so the first attempt 
at a systematic presentation of our science. We cannot here do 
more than glance over the mass of rules contained in this work, 
giving only its purpose, a discussion of the analysis of statements 
as made by Aristotle in this early work, and a brief review of his 
teaching about fallacies. The most important of the formal rules 
and laws of inference occurring here, continued, as it seems, to be 
recognized as valid in the later works, and they will therefore be 
considered in the section on the non-analytical formulae (§ 16). 

49 



THE GREEK VARIETY OF LOGIC 

11.01 Next in order after the foregoing, we must say for 
how many and for what purposes the treatise is useful. They 
are three - intellectual training, casual encounters, and the 
philosophical sciences. That it is useful as a training is obvious 
on the face of it. The possession of a plan of inquiry will 
enable us more easily to argue about the subject proposed. 
For purposes of casual encounters, it is useful because when 
we have counted up the opinions held by most people, we shall 
meet them on the ground not of other people's convictions 
but of their own, while we shift the ground of any argument 
that they appear to us to state unsoundly. For the study of 
the philosophical sciences it is useful, because the ability to 
raise searching difficulties on both sides of a subject will make 
us detect more easily the truth and arror about the several 
points that arise. It has a further use in relation to the 
ultimate bases of the principles used in the several sciences. 
For it is impossible to discuss them at all from the principles 
proper to the particular science in hand, seeing that the prin- 
ciples are the prius of everything else : it is through the 
opinions generally held on the particular points that these 
have to be discussed, and this task belongs properly, or most 
appropriately, to dialectic: for dialectic is a process of criti- 
cism wherein lies the path to the principles of all inquiries. 

The logic thus delineated treats of propositions and problems, 
described as follows: 

11.02 The materials with which arguments start are equal 
in number, and are identical, with the subjects on which 
reasonings take place. For arguments start with 'propositions', 
while the subjects on which reasonings take place are 'pro- 
blems'. 

11.03 The difference between a problem and a proposition 
is a difference in the turn of the phrase. For if it be put in 
this way, ' "An animal that walks on two feet" is the definition 
of man, is it not ?' or ' "Animal" is the genus of man, is it not ?' 
the result is a proposition: but if thus, 'Is "an animal that 
walks on two feet" a definition of man or no?' the result is a 
problem. Similarly too in other cases. Naturally, then, pro- 
blems and propositions are equal in number: for out of every 
proposition you will make a problem if you change the turn 
of the phrase. 

50 



ARISTOTLE 

Of epoch-making importance is the classification of methods of 
proof given in the same connection : 

11.04 Having drawn these definitions, we must distinguish 
how many species there are of dialectical arguments. There 
is on the one hand Induction, on the other Syllogism. Now 
what a syllogism is has been said before: induction is a 
passage from individuals to universals, e.g. the argument that 
supposing the skilled pilot is the most effective, and likewise 
the skilled charioteer, then in general the skilled man is the 
best at his particular task. Induction is the more convinf ring 
and clear: it is more readily learnt by the use of the senses, 
and is applicable generally to the mass of men, though 
syllogism is more forcible and effective against contradictious 
people. 

The subject of the Topics are essentially the so-called loci (totcoi). 
Aristotle never defined them, and so far no-one has succeeded in 
saying briefly and clearly what they are. In any case it is a matter 
of certain very general prescriptions for shaping arguments. 

An example: 

11.05 Now one commonplace rule (totzoc,) is to look and see 
if a man has ascribed as an accident what belongs in some other 
way. This mistake is most commonly made in regard to the 
genera of things, e.g. if one were to say that white happens to 
be a colour - for being a colour does not happen by accident to 
white, but colour is its genus. 

B. PREDICABLES 

As an introduction to these loci Aristotle in the first book of the 
Topics developed two different doctrines of the structure of state- 
ments, both of which obtained considerable historical importance 
and still remain of interest: namely the doctrines of the so-called 
predicables and of the categories. 

11.06 Every proposition and every problem indicates 
either a genus or a peculiarity or an accident - for the dif- 
ferentia too, applying as it does to a class (or genus), should 
be ranked together with the genus. Since, however, of what 
is peculiar to anything part signifies its essence, while part 
does not, let us divide the 'peculiar' into both the aforesaid 

51 



THE GREEK VARIETY OF LOGIC 

parts, and call that part which indicates the essence a 'defi- 
nition', while of the remainder let us adopt the terminology 
which is generally current about these things, and speak of 
it as a 'property'. 

11.07 We must now say what are 'definition', 'property', 
'genus', and 'accident'. A 'definition' is a phrase signifying a 
thing's essence. It is rendered in the form either of a phrase 
in lieu of a term, or of a phrase in lieu of another phrase; for 
it is sometimes possible to define the meaning of a phrase as 
well. 

11.08 A 'property' (iSiov) is a predicate which does not 
indicate the essence of a thing, but yet belongs to that thing 
alone, and is predicated convertibly of it. Thus it is a property 
of man to be capable of learning grammar: for if A be a man, 
then he is capable of learning grammar, and if he be capable 
of learning grammar, he is a man. 

11.09 A 'genus' is what is predicated in the category of 
essence of a number of things exhibiting differences in kind. 

11.10 An 'accident' is (1) something which though it is 
none of the foregoing - i.e. neither a definition nor a property 
nor a genus - yet belongs to the thing: (2) something which 
may possibly either belong or not belong to any one and the 
self-same thing, as (e.g.) the 'sitting posture' may belong or 
not belong to some self-same thing. 

The logical significance of this division of the 'predicates' 
consists in the fact that it is an attempt to analyse propositions, with 
reference moreover to the relation between subject and predicate. 
This analysis is effected in terms of the matter rather than the form, 
yet contains echoes of purely structural considerations, as for instance 
in the distinction between genus and specific difference or property, 
where the genus is evidently symbolized by a name, properties by a 
functor. 

As a kind of pendent to the doctrine of the predicables, Aristotle 
presents a theory of identity: 

11.11 Sameness would be generally regarded as falling, 
roughly speaking, into three divisions. We generally apply the 
term numerically or specifically or generically - numerically 
in cases where there is more than one name but only one 
thing, e.g. 'doublet' and 'cloak'; specifically, where there is 
more than one thing, but they present no differences in respect 
of their species, as one man and another, or one horse and 

52 



ARISTOTLE 



another: for things like this that fall under the same species 
are said to be 'specifically the same'. Similarly, too, those 
things are called generically the same which fall under the 
same genus, such as a horse and a man. 



C. CATEGORIES 

Another analysis of propositions is contained in the theory of the 
categories. This seems to be a systematic development of hints in 
Plato. Only in one place (apart from the Categories: 11.12) do we 
find an enumeration of ten categories (the only one usually ascribed 
to Aristotle): 

11.13 Next, then, we must distinguish the classes of 
predicates in which the four orders in question (11.06 — 11.10) 
are found. These are ten in number: Essence, Quantity, Quali- 
ty, Relation, Place, Time, Position, State, Activity, Passivity. 
For the accident and genus and property and definition of 
anything will always be in one of these categories : for all the 
propositions found through these signify either something's 
essence or its quality or quantity or some one of the other 
types of predicate. It is clear, too, on the face of it that the 
man who signifies something's essence signifies sometimes a 
substance, sometimes a quality, sometimes some one of the 
other types of predicate. For when a man is set before him 
and he says that what is set there is 'a man' or 'an animal', 
he states its essence and signifies a substance; but when a 
white colour is set before him and he says that what is set 
there is 'white' or is 'a colour', he states its essence and signi- 
fies a quality. Likewise, also, if a magnitude of a cubit be 
set before him and he says that what is set there is a magnitude 
of a cubit, he will be describing its essence and signifying a 
quantity. Likewise also in the other cases. 

This text contains an ambiguity: 'essence' (xi kaxi) means first 
a particular category - that of substance (oucria) as we see from a 
parallel text of the Categories (11.14) - secondly that essence or 
intrinsic nature which is found in every category, not only in that of 
substance. The thought becomes clear if 'substance' is put in the 
list of the ten categories in place of 'essence'. 

Here the doctrine of the categories is treated as a division of 
sentences and problems for practical purposes. But beyond this 
Aristotle regarded it as involving two more important problems. 
In the Prior Analytics we read : 

53 



THE GREEK VARIETY OF LOGIC 

11.15 The expressions 'this belongs to that' and 'this holds 
true of that' must be understood in as many ways as there 
are different categories. 

That means that the so-called copula of the sentence has as many 
meanings as there are categories. That is the first reason why the 
theory of the categories is logically so important. The second is 
that while this theory constitutes an attempt at classifying objects 
according to the ways in which they are predicable, it put in Aristo- 
tle's path the problem of the univeral class. He solved it with 
brilliant intuition, though, as we now know, with the help of a 
faulty proof. The relevant passage occurs in the third book of the 
Metaphysics: 

11.16 It is not possible that either unity or being should 
be a single genus of things; for the differentiae of any genus 
must each of them both have being and be one, but it is not 
possible for the genus taken apart from its species (any more 
than for the species of the genus) to be predicated of its proper 
differentiae; so that if unity or being is a genus, no differentia 
will either have being or be one. 

The line of thought which Aristotle expresses in this very com- 
pressed formula is as follows: 

1. For all A: if A is a genus, then there is (at least) one B, which 
is the specific difference of A; 

2. for all A and B: if B is the specific difference of A, then not: 
B is A. Suppose now 

3. there is an all-inclusive genus V: of this it would be true that 

4. for every B: B is V. 

As V is a genus, it must have a difference (by 1.); call it D. Of this 
D it would be true on the one hand that D is V (by 4.), and on the 
other that D is not V (by 2.). Thus a contradiction results, and 
at least one of the premisses must be false (cf. 16.33). As Aristotle 
holds 1. and 2. to be true, he must therefore reject the supposition 
that there is an all-inclusive genus (3.): there is no summum genus. 
We have here the basis of the scholastic doctrine of analogy (28.19) 
and the first germ of a theory of types (cf. § 47). 

The proof is faulty : for 'D is V is not false but meaningless (48.24). 
But beyond all doubt the thought confronting us deserves to be 
styled a brilliant intuition. 

D. SOPHISTIC 

The last book of the Topics, known as the Sophistic Befutations, 
contains an extensive doctrine of fallacious inferences. Like most 
other parts of the Topics this one too belongs to the first form of 
Aristotelian logic, not yet formal, but guided by the practical 

54 



ARISTOTLE 

interests of every-day discussion. There is a second doctrine of 
fallacious inference, in the Prior Analytics (11.17j, much briefer 
than the first but incomparably more formal; all fallacious inferences 
are there reduced to breaches of syllogistic laws. However neither 
Aristotle himself, nor anyone after him, really succeeded in replacing 
the doctrine of the Sophistic Refutations, primitive though it is 
from the formal point of view. Knowledge of it is also indispensable 
for the understanding of scholastic logic. For all of which reasons 
we shall cite a few passages from it here. 

11.18 Refutation is reasoning involving the contradictory 
of the given conclusion. Now some of them do not really 
achieve this, though they seem to do so for a number of 
reasons; and of these the most prolific and usual domain is 
the argument that turns upon names only. It is impossible in 
a discussion to bring in the actual things discussed : we use 
their names as symbols instead of them; and therefore we 
suppose that what follows in the names, follows in the things 
as well, just as people who calculate suppose in regard to 
their counters. But the two cases (names and things) are not 
alike. For names are finite and so is the sum-total of formulae, 
while things are infinite in number. Inevitably, then, the same 
formulae, and a single name, have a number of meanings. 

Historically, a very important text: in it Aristotle rejects forma- 
lism, rightly so for the purposes of ordinary language. For without 
preliminary distinction of the various functioning of signs correct 
laws cannot be formulated in such a language. The text just cited 
underlies the vast growth of medieval doctrine about supposition, 
appellation and analogy (§§ 27 and 28). So far as concerns Aristotle 
and the other ancient logicians, it appears that they got round the 
difficulty mentioned by applying rules by which ordinary language 
was turned into an artificial language with a single function for 
every verbal form. 

11.19 There are two styles of refutation: for some depend 
on the language used, while some are independent of language. 
Those ways of producing the false appearance of an argument 
which depend on language are six in number: they are 
ambiguity, amphiboly, combination, division of words, 
accent, form of expression. 

11.20 Arguments such as the following depend upon ambi- 
guity. 'Those learn who know: for it is those who know their 
letters who learn the letters dictated to them.' For 'to learn' 

55 



THE GREEK VARIETY OF LOGIC 

is ambiguous; it signifies both 'to understand' by the use 
of knowledge, and also 'to acquire knowledge'. 

11.21 Examples such as the following depend upon amphi- 
boly: . . . 'Speaking of the silent is possible': for 'speaking of 
the silent' also has a double meaning: it may mean that the 
speaker is silent or that the things of which he speaks are so. 

11.22 Upon the combination of words there depend 
instances such as the following: 'A man can walk while 
sitting, and can write while not writing'. For the meaning is 
not the same if one divides the words and if one combines 
them in saying that 'it is possible to walk-while-sitting . . .'. 
The same applies to the latter phrase, too, if one combines the 
words 'to write-while-not-writing' : for then it means that he 
has the power to write and not to write at once; whereas if 
one does not combine them, it means that when he is not 
writing he has the power to write. 

11.23 Upon division depend the propositions that 5 is 2 
and 3, and even and odd, and that the greater is equal: for 
it is that amount and more besides. 

11.24 Of fallacies, on the other hand, that are independent 
of language there are seven kinds : 

(1) that which depends upon Accident: 

(2) the use of an expression absolutely or not absolutely but 
with some qualification of respect, or place, or time, or 
relation : 

(3) that which depends upon ignorance of what 'refutation' 
is: 

(4) that which depends upon the consequent: 

(5) that which depends upon assuming the original con- 
clusion : 

(6) stating as cause what is not the cause: 

(7) the making of more than one question into one. 

And example of (1) is: 'If Coriscus is different from a man he is 
different from himself (11.25); of (2): 'Suppose an Indian to be 
black all over, but white in respect of his teeth ; then he is both white 
and not white' (11.26); (3) consists in proving something other 
than what is to be proved (11.27); (5) consists in presupposing what 
is to be proved (11.28). (4) alone involves a formal fallacy, namely 
concluding from the consequent to the antecedent of a conditional 
sentence (11.29). 



56 



§12. THEORY OF OPPOSITION; 
PRINCIPLE OF CONTRADICTION; 
PRINCIPLE OF TERTIUM EXCLUSUM 

A. THEORY OF OPPOSITION 

Aristotle developed two different theories of opposition. The 
first, contained in the Topics (12.01) and belonging to the earlier 
period of his development, is most clearly summarized in the 
pseudo-aristotelian Categories : 

12.02 There are four senses in which one thing is said to be 
opposed to another: as correlatives, or as contraries, or as 
privation and habit (e?i?), or as affirmation and denial. To 
give a general outline of these oppositions: the double is 
correlative to the half, evil is contrary to good, blindness is a 
privation and sight a habit, 'he sits' is an affirmation, 'he does 
not sit' a denial. 

Two points are worth remark: the division presupposes a material 
standpoint, and even in the last case, contradictory opposition, 
concerns relationships between terms, not sentences. 

It is quite otherwise in the later period. The second doctrine 
presupposes the Aristotelian theory of quantification, which is later 
than the Topics: 

12.03 Some things are universal, others individual. By the 
term 'universal' I mean that which is of such a nature as to be 
predicated of many subjects, by 'individual' that which is not 
thus predicated. Thus 'man' is a universal, 'Callias' an indi- 
vidual. . . . If, N then, a man states a positive and a negative 
proposition of universal character with regard to a universal, 
these two propositions are 'contrary'. By the expression 'a 
proposition of universal character with regard to a universal', 
such propositions as 'every man is white', 'no man is white' 
are meant. When, on the other hand, the positive and negative 
propositions, though they have regard to a universal, are yet 
not of universal character, they will not be contrary, albeit the 
meaning intended is sometimes contrary. As instances of 
propositions made with regard to a universal, but not of 
universal character, we may take the propositions 'man is 
white', 'man is not white'. 'Man 'is a universal, but the pro- 
position is not made as of universal character; for the word 
'every' does not make the subject a universal, but rather 
gives the proposition a universal character. If, however, both 

57 



THE GREEK VARIETY OF LOGIC 

predicate and subject are distributed, the proposition thus 
constituted is contrary to truth; no affirmation will, under 
such circumstances, be true. The proposition 'every man is 
every animal' is an example of this type. 

This text contains the following points of doctrine: 1. distinction 
between general and singular sentences, according to the kind of 
subject; 2. divison of general sentences into universal and particular 
according to the extension of the subject; 3. rejection of quantifica- 
tion of the predicate. The whole doctrine is now purely formal, and 
is explicitly concerned with sentences. 

Another division is to be found at the beginning of the Prior 
Analytics: 

12.04 A premiss then is a sentence affirming or denying one 
thing of another. This is either universal or particular or 
indefinite. By universal I mean the statement that something 
belongs to all or none of something else; by particular that it 
belongs to some or not to some or not to all; by indefinite that 
it does or does not belong, without any mark to show whether 
it is universal or particular. 

Here Aristotle enumerates three kinds of sentence: universal, 
particular and indefinite. It is striking that no mention is made of 
singular sentences. This is due to the fact that every term in the 
syllogistic must be available both as subject and predicate, but accord- 
ing to Aristotle singular terms cannot be predicated (12.05). In the 
particular sentence 'some' means 'at least one, not excluding all'. 
Whereas, as Sugihara has recently shewn (12.06), an indefinite 
sentence should probably be interpreted in the sense: 'at least one 
A is B and at least one A is not B\ However cases in which the 
formal properties of particular and indefinite sentences differ are 
rare in the syllogistic, so that Aristotle himself often states the 
equivalence of these sentences (12.07). Later, even in Alexander of 
Aphrodisias (12.08), these cases are dropped altogether. 

12.09 Verbally four kinds of opposition are possible, viz. 
universal affirmative to universal negative, universal affirma- 
tive to particular negative, particular affirmative to universal 
negative, and particular affirmative to particular negative: 
but really there are only three : for the particular affirmative is 
only verbally opposed to the particular negative. Of the 
genuine opposites I call those which are universal contraries, 
the universal affirmative and the universal negative, e.g. 'all 
science is good', 'no science is good'; the others I call contra- 
dictories. 

58 



ARISTOTLE 

Here we have the 'logical square' later to become classical, which 
can be set out schematically thus: 

12.091 

to belong to all contrary to belong to none 



\ 



/ 



% .if* 

V 

/ * . 
/ \ 

to belong to some only verbal not to belong to all 

The logical relationships here intended are shown in the following 
passages: 

12.10 Of such corresponding positive and negative pro- 
positions as refer to universals and have a universal character, 
one must be true and the other false. 

12.11 It is evident also that the denial corresponding to a 
single affirmation is itself single; ... for instance, the affir- 
mation 'Socrates is white' has its proper denial in the pro- 
position 'Socrates is not white' . . . The denial proper to the 
affirmation 'every man is white' is 'not every man is white'; 
that proper to the affirmation 'some man is white' is 'no man 
is white'. 

In the later tradition the so-called laws of subalternation also 
came to be included in the 'logical square'. They run: 
If A belongs to all B, then it belongs to some B (12.12). 
If A belongs to no B, then to some B it does not belong (12.13). 

B. OBVERSION 

In Aristotle's logic negation normally occurs only as a functor 
determining a sentence, but there are a number of places in the 
Organon where formulae are considered which contain a negation 
determining a name. Thus we read in the Hermeneia: 

12.14 The proposition 'no man is just' follows from the 
proposition 'every man is not-just' and the proposition 'not 
every man is not-just', which is the contradictory of 'every 

59 



THE GREEK VARIETY OF LOGIC 

man is not-just', follows from the proposition 'some man is 
just'; for if this be true, there must be some just man. 

These laws were evidently discovered with great labour and after 
experimenting with various false formulae (12.15). Aristotle has 
similar ones for sentences with individual subjects in the Hermeneia 
(12.16). 

In the Prior Analytics Aristotle develops a similar doctrine in 
more systematic form and with variables: 

12.17 Let A stand for 'to be good', B for 'not to be good', 
let C stand for 'to be not-good', and be placed under B, and 
let D stand for 'not to be not-good' and be placed under A. 
Then either A or B will belong to everything, but they will 
never belong to the same thing; and either C or D will belong 
to everything, but they will never belong to the same thing. 
And B must belong to everything to which C belongs. For if 
it is true to say 'it is not-white', it is true also to say 'it is not 
white' : for it is impossible that a thing should simultaneously 
be white and be not-white, or be a not-white log and be a 
white log; consequently if the affirmation does not belong, 
the denial must belong. But C does not always belong to B: 
for what is not a log at all, cannot be a not-white log either. 
On the other hand D belongs to everything to which A belongs. 
For either C or D belongs to everything to which A belongs. 
But since a thing cannot be simultaneously not-white and 
white, D must belong to everything to which A belongs. For 
of that which is white it is true to say that it is not not-white. 
But A is not true of all D. For of that which is not a log at all 
it is not true to say A, viz. that it is a white log. Consequently 
D is true, but A is not true, i.e. that it is a white log. It is clear 
also that A and C cannot together belong to the same thing, 
and that B and D may possibly belong to the same thing. 



C. THE PRINCIPLE OF CONTRADICTION 

While Aristotle was well acquainted with the principle of identity, 
so much discussed later, he only mentions it in passing (12.18). But 
to the principle of contradiction he devoted the whole of Book T 
of the Metaphysics. This book is evidently a youthful work, and was 
perhaps written in a state of excitement, since it contains logical 
errors; nevertheless it is concerned with an intuition of fundamental 
importance for logic. 

60 



ARISTOTLE 

The following are the most important formulations of the principle 
of contradiction: 

12.19 The same attribute cannot at the same time belong 
and not belong to the same subject and in the same respect. 

12.20 Let A stand for 'to be good', B for 'not to be good' .... 
Then either A or B will belong to everything, but they will 
never belong to the same thing. 

12.21 It is impossible that contradictories should be at the 
same time true of the same thing. 

12.22 It is impossible to affirm and deny truly at the same 
time. 

The first two of these formulae are in the object language, the 
last two in a metalanguage, and the author evidently understands 
the difference. 

In the Topics and Hermeneia Aristotle has a stronger law: 

12.23 It is impossible that contrary predicates should belong 
at the same time to the same thing. 

12.24 Propositions are opposed as contraries when both 
the affirmation and the denial are universal ... in a pair of 
this sort both propositions cannot be true. 

This last statement is quite understandable if one remembers 
that when Aristotle was young proofs principally consisted of 
refutations. But when Aristotle had developed his syllogistic, in 
which refutation has only a subordinate part to play, he not only 
found that the principle of contradiction would not do at all as the 
first axiom, but also that violence may be done to it in a correct 
syllogism. 

The first in modern times to advert to this Aristotelian doctrine 
was I. Husic in 1906 (12.27). It may seem so astonishing to readers 
accustomed to the 'classical' interpretation of Aristotelian logic, 
that it is worth while shewing not merely its absolute necessity 
but also the context from which our logician's thought clearly 
emerges. 

12.28 The law that it is impossible to affirm and deny 
simultaneously the same predicate of the same subject is not 
expressly posited by any demonstration except when the 
conclusion also has to be expressed in that form; in which case 
the proof lays down as its major premiss that the major 
is truly affirmed of the middle but falsely denied. It makes no 
difference, however, if we add to the middle, or again to the 

61 



THE GREEK VARIETY OF LOGIC 

minor term, the corresponding negative. For grant a minor 
term of which it is true to predicate man - even if it be also 
true to predicate not-man of it -still grant simply that man is 
animal and not not-animal, and the conclusion follows: for 
it will still be true to say that Callias - even if it be also true 
to say that not-Callias - is animal and not not-animal. 

The syllogism here employed has then, omitting quantifiers, 
the following form: 

12.281 If M is P and not not-P, and £ is M, then S is P 
and not not-P. 

So the principle of contradiction is no axiom, and does not need 
to be presupposed, except in syllogisms of the fore-going kind. 
The text quoted is also remarkable in that the middle term in 12.281 
is a product (cf. the commentary on 15.151), and that in the minor 
term an individual name is substituted (cf. § 13, C, 5) - in each 
case contrary to normal syllogistic practice. However the text 
comes from the Posterior Analytics and must belong to a relatively 
early period. 

Aristotle goes still further and states that the principle of con- 
tradiction can be completely violated in a conclusive syllogism: 

12.29 In the middle (i.e. second: cf. 13.07) figure a syllo- 
gism can be made both of contradictories and contraries. Let 
'A' stand for 'good', let ' B' and 'C stand for 'science'. If then 
one assumes that every science is good, and no science is good, 
A belongs to all B and to no C, so that B belongs to no C: 
no science is then a science. 

This syllogism has the following form : 

12.291 If all M is P and no M is P, then no M is M. 

D. THE PRINCIPLE OF TERTIUM EXCLUSUM 

One formulation of this principle has already been quoted (12.20). 
Others are: 

12.30 In the case of that which is, or which has taken place, 
propositions, whether positive or negative, must be true or 
false. Again, in the case of a pair of contradictories, either when 
the subject is universal and the propositions are of a universal 
character, or when it is individual, as has been said, one of the 
two must be true and the other false. 

62 



ARISTOTLE 

12.31 One side of the contradiction must be true. Again, if 
it is necessary with regard to everything either to assert or to 
deny it, it is impossible that both should be false. 

Aristotle's normal practice was to presuppose the correctness of 
these theses, and he devoted a notable chapter of the fourth hook of 
the Metaphysics (T 8) to the defence of the principle of tertium 
exclusion (or tertium non dalur). At least once, however, he called it 
into question: in the ninth chapter of the Hermeneia he will not 
allow it to be valid for future contingent events. He bases his ar- 
gument thus: 

12.32 If it is true to say that a thing is white, it must 
necessarily be white; if the reverse proposition is true, it 
will of necessity not be white .... It may therefore be argued 
that it is necessary that affirmations or denials must be 
either true or false. Now if this be so, nothing is or takes 
place fortuitously, either in the present or in the future, and 
there are no real alternatives; everthing takes place of neces- 
sity and is fixed. ... It is therefore plain that it is not necessary 
that of an affirmation and a denial one should be true and the 
other false. For in the case of that which exists potentially, 
but not actually, the rule which applies to that which exists 
actually does not hold good. 

These considerations had no influence on Aristotle's logical 
system, as has already been said, but they came to have great 
historical importance in the Middle Ages. 

The doubt about the validity ot the principle of tertium exclusum 
arose from an intuition of the difficult problems which it sets. The 
debate is not closed even today. 



§13. ASSERTORIC SYLLOGISTIC 

We give here a page of the Prior Analytics in as literal a trans- 
lation as possible, and comment on it afterwards. It contains the 
essentials of what later came to be called Aristotle's assertoric 
syllogistic. It is a leading text, in which no less than three great 
discoveries are applied for the first time in history: variables, purely 
formal treatment, and an axiomatic system. It constitutes the 
beginning of formal logic. Short as it is, it formed the basis of logical 
speculation for more than two thousand years - and yet has been 
only too often much misunderstood. It deserves to be read atten- 
tively. 

63 



THE GREEK VARIETY OF LOGIC 
A. TEXT 

Aristotle begins by stating the laws of conversion of sentences. 
These are cited below, 14.10 ff., among the bases of the systematic 
development. He goes on : 

13.01 When then three terms are so related one to another 
that the last is in the middle (as in a) whole and the middle is or is 
not in the first as in a whole, then there must be a perfect 
syllogism of the two extremes. 

13.02 Since if A (is predicated) of all B, and B of all C, A 
must be predicated of all C. 

13.03 Similarly too if A (is predicated) of no B, and B of 
all A, it is necessary that A will belong to no C. 

13.04 But if the first follows on all the middle whereas the 
middle belongs to none of the last, there is no syllogism of the 
extremes; for nothing necessary results from these; for the 
first may belong to all and to none of the last; so that neither 
a particular nor a universal is necessary; and since there is 
nothing necessary these produce no syllogism. Terms for 
belonging to all: animal, man, horse; for belonging to none: 
animal, man, stone. 

13.05 But if one of the terms is related wholly, one partially, 
to the remaining one; when the wholly related one is posited 
either affirmatively or negatively to the major extreme, and 
the partially related one affirmatively to the minor extreme, 
there must be a perfect syllogism . . . for let A belong to all B 
and B to some C, then if being predicated of all is what has 
been said, A must belong to some C. 

13.06 And if A belongs to no B and B to some C, to some 
C A must not belong . . . and similarly if the BC (premiss) is 
indefinite and affirmative. 

13.07 But when the some belongs to all of one, to none of 
the other; such a figure I call the second. 

13.08 For let M be predicated of no N and of all X ; since 
then the negative converts, N will belong to no M ; but M 
was assumed (to belong) to all X ; so that N (will belong) to 
no X ; for this has been shewn above. 

13.09 Again, if M (belongs) to all iV and to no X, X too 
will belong to no N; for if M (belongs) to no X, X too (belongs) 
to no M ; but M belonged to all N ; therefore X will belong to 
no iV, for the first figure has arisen again ; but since the nega- 
tive converts, N too will belong to no X, so that it will be the 

64 



ARISTOTLE 

same syllogism. It is possible to shew this by bringing to 
impossibility. 

13.10 If M belongs to no TV and to some X, then to some 
X TV must not belong. For since the negative converts, TV 
will belong to no M, but M has been supposed to belong to 
some X; so that to some X TV will not belong; for a syllogism 
arises in the first figure. 

13.11 Again, if M belongs to all TV and to some X not. to 
some TV X must not belong. For if it belongs to all, and M is 
predicated of all TV, M must belong to all X ; but to some it has 
been supposed not to belong. 

13.12 If to the same, one belongs to all, the other to none, or 
both to all or to none, such a figure I call third. 

13.13 When both P and R belong to all S, of necessity P 
will belong to some B; for since the affirmative converts, S 
will belong to some P, so that when P (belongs) to all S, and 
S to some R, P must belong to some R; for a syllogism 
arises in the first figure. One can make the proof also by (bring- 
ing to) the impossible and by setting out (terms); for if both 
belong to all S, if some of the (things which are) S be taken, say TV, 
to this both P and R will belong, so that to some R P will belong. 

13.14 And if R belongs to all S, and P to none, there will 
be a syllogism that to some R P necessarily will not belong. 
For there is the same manner of proof, with the RS premiss 
converted. It could also be shewn by the impossible as in the 
previous cases. 

13.15 If R (belongs) to all S and P to some, P must belong 
to some R. For since the affirmative converts, 8 will belong 
to some P, so that when R (belongs) to all S, and S to some P, 
R too will belong to some P; so that P (will belong) to some R. 

13.16 Again, if R belongs to some S and P to all, P must 
belong to some R; for there is the same manner of proof. 
One can also prove it by the impossible and by setting out, 
as in the previous cases. 

13.17 If R belongs to all S, and to some (S) P does not. 
then to some R P must not belong. For if to all, and R (belongs) 
to all S, P will also belong to all S; but it did not belong. It is 
also proved with reduction (to the impossible) if some of 
what is S be taken to which P does not belong. 

13.18 If P belongs to no S, and R to some S, to some R P 
will not belong; for again there will be the first figure when the 
RS premiss is converted. 

65 



THE GREEK VARIETY OF LOGIC 
B. INTERPRETATION 

This passage is composed in such compressed language that most 
readers find it very hard to understand. Indeed the very style is of 
the greatest significance for the history of logic ; for here we have the 
manner of thought and writing of all genuine formal logicians, be 
they Stoics or Scholastics, be their name Leibniz or Frege. Hence 
we have given a literal version, but shall now interpret it with the 
aid of paraphrase and commentary: 

on 13.01 : Aristotle defines the first figure. This may serve as an 
example: Gainful art is contained in art in general as in a whole; 
the art of pursuit (e.g. hunting) is contained in gainful art as in 
a whole; therefore the art of pursuit is contained in art in general 
as in a whole. The example is taken from Plato's division (8.05), 
from which the Aristotelian syllogism seems to have developed. 

We shall explain what a perfect syllogism is in § 14. 

on 13.02: This mood later (with Peter of Spain) came to be called 
'Barbara'. Hereafter we refer to all moods by the mnemonic names 
originating with Peter of Spain (cf. 32.04 ff.). 

We obtain an example by substitution: 
If animal belongs to all man 
and man belongs to all Greek, 
then animal belongs to all Greek. 

on 13.03: Celarent: If stone belongs to no man 

and man belongs to all Greek, 
then stone belongs to no Greek. 

on 13.04: Names are here given for two substitutions by which 
it can be shewn that a further mood is invalid. Probably the follow- 
ing are intended : 

Mood Substitution 

If A to all B 1. If animal belongs to all man 

and B to no C, and man belongs to no horse, 

then A to no C, then animal belongs to no horse. 

2. If animal belongs to all man 
and man belongs to no stone, 
then animal belongs to no stone. 
In each case the premisses are true, but the conclusion is once 
true, once false. Therefore the mood is invalid. 

on 13.05: Darii: If Greek belongs to all Athenian 

and Athenian belongs to some logician, 
then Greek belongs to some logician. 
It is to be noticed that here and subsequently 'some' must have 
the sense of 'at least one'. 

on 13.06: Ferio: If Egyptian belongs to no Greek 

and Greek belongs to some logician, 

then to some logician Egyptian does not belong. 

66 



ARISTOTLE 

on 13.07: Aristotle defines the second figure, in which the middle 
term is predicate in both premisses. He considers three cases: 1. one 
premiss is universal and affirmative, the other universal and negative, 
2. both premisses are universal and affirmative, '.>. both premisses 

are negative. Only in the first case there are valid syllogisms, 
on 13.08: Cesare: If man belongs to no stone 

and man belongs to all Greek, 
then stone belongs to no Greek. 
This is reduced to Celarent (13.03) by conversion of the major 
(first) premiss: If stone belongs to no man 

and man belongs to all Greek, 
then stone belongs to no Greek, 
on 13.09: Camestres: If (1) animal belongs to all man 

and (2) animal belongs to no stone, 
then (3) man belongs to no stone. 
The proof proceeds by reduction to Celarent (13.03). First the minor 
premiss (2) is converted : 

(4) stone belongs to no animal; 
then comes the other premiss : 

(1) animal belongs to all man. 
(4) and (1) are the premisses of Celarent, from which follows the 
conclusion, 

(5) stone belongs to no man. This is converted, and so the desired 
conclusion is obtained. 

It is important to notice that this conclusion is first stated by 
Aristotle at the end of the process of proof. 

on 13.10: Festino: The proof of this mood is by reduction to Ferio 
(13.06), the major premiss being converted as in Cesare (13.08). 
on 13.11: Baroco: If (1) Greek belongs to all Athenian 

and (2) to some logician Greek does not belong, 
then (3) to some logician Athenian does not 
belong. 
The proof proceeds by first hypothesizing the conclusion as false, 
i.e. its contradictory opposite is supposed : 

(4) Athenian belongs to all logician. 
Now comes the first (major) premiss: 

(5) Greek belongs to all Athenian. 
From these one obtains a syllogism in Barbara (13.02): 

If (5) Greek belongs to all Athenian 
and (4) Athenian belongs to all logician, 
then (6) Greek belongs to all logician. 
But the conclusion (6) of this syllogism is contradictorily opposed 
to the minor premiss of Baroco, (2), and as this is accepted, the former 
must be rejected. So one of the premisses (4) and (5) must be rejected : 
as (5) is accepted, (4) must be rejected; and so we obtain the con- 
tradictory opposite of (4), namely (3). 

67 



THE GREEK VARIETY OF LOGIC 

on 13.12: Aristotle defines the third figure, in which the middle 
term is subject in both premisses. He considers the same three 
cases as in 13.07. 

on 13.13: Darapti: If Greek belongs to all Athenian 
and man belongs to all Athenian, 
then Greek belongs to some man. 
This syllogism is first reduced to Darii (13.05) by conversion of the 
minor premiss - just as Cesare (13.08) is reduced to Celarent (13.03). 
But Aristotle then develops two further methods of proof: a process 
- as with Baroco (13.11) - 'through the impossible', and the 'setting 
out of terms'. This last consists in singling out a part, perhaps an 
individual (but this is debated in the literature), from the Athenians, 
say Socrates. It results that as Greek as well as man belongs to all 
Athenian, Socrates must be Greek as well as man. Therefore this is 
a Greek who is man. Accordingly Greek belongs to (at least) one 
man. 

on 3.14: Felapton: If Egyptian belongs to no Athenian 
and man belongs to all Athenian, 
then to some man Egyptian does not belong. 
The proof proceeds by conversion of the minor (second) premiss, 
resulting in Ferio (13.06). 

on 3.15: Disamis: If Athenian belongs to all Greek 

and logician belongs to some Greek, 
then logician belongs to some Athenian. 
The first thing to be noticed is that Aristotle here writes the minor 
premiss first, as also in 13.16 and 13.17. The proof is by reduction 
to Darii (13.05), just as Camestres (13.09) was reduced to Celarent 
(13.03). 

on 13.16: Datisi: If logician belongs to some Greek 
and Athenian belongs to all Greek, 
then Athenian belongs to some logician. 
The proof proceeds by reduction to Darii (13.05), the minor premiss 
(here the first!) being converted. 

on 13.17: Bocardo: If Greek belongs to all Athenian 

and to some Athenian logician does not 

belong, 
then to some Greek logician does not belong. 
The premisses are again in reversed order, the minor coming first. 
The proof proceeds by reduction to the impossible, with use of 
Barbara (13.02) as in the case of Baroco (13.11). A proof by setting 
out of terms is also recommended, but not carried through, 
on 13.18: Ferio: If Egyptian belongs to no Greek 

and logician belongs to some Greek, 
then to some logician Egyptian does not belong. 
Aristotle reduces this syllogism to Ferio (13.06), by conversion of 
the minor (second) premiss. 

68 



ARISTOTLE 

This paraphrase with comments is, be it noted, a concession to the 
modern reader. For in his Analytics Aristotle never argued by means 
of substitutions, as we have been doing, except in proofs of invalidity. 
However, in view of the contemporary state of logical awareness, it 
seems necessary to elucidate the text in this more elementary way. 



C STRUCTURE OF THE SYLLOGISM 

If we consider the passages on which we have just commented, 
the first thing we notice is that the definition which Aristotle gives of 
the syllogism (10.05), does indeed contain it, but is much too wide: 
the analytic syllogism as we call the class of formulae considered in 
chapters 4-6 of the first book of the Prior Analytics, can be described 
as follows: 

1. It is a conditional sentence, the antecedent of which is a con- 
junction of two premisses. Its general form is : ' If p and q, then r', in 
which propositional forms are to be substituted for 'p', 'g' and V. So 
the Aristotelian syllogism has not the later form: 'p; q; therefore r', 
which is a rule. The Aristotelian syllogism is not a rule but a proposition. 

2. The three propositional forms whose inter-connection produces a 
syllogism, are always of one of these four kinds: 'A belongs to all B\ 

A belongs to no B\ l A belongs to (at least) some B\ ' (at least) to 
some B A does not belong'. Instead of this last formula, there some- 
times occurs the (equivalent) one: 'A belongs not to all B\ The 
word 'necessary' or 'must' is often used : evidently that only means 
here that the conclusion in question follows logically from the 
premisses. 

3. Where we have been speaking not of propositions but of 
forms, seeing that Aristotelian syllogisms always contain letters 
('A', 'B', T' etc.) in place of words, which are evidently to be 
interpreted as variables, Aristotle himself gives examples of how 
substitutions can be made in them. That is indeed the only kind 
of substitution for variables known to him: he has, for example, 
no thought of substituting variables for variables. Nevertheless 
this is an immense discovery : the use of letters instead of constant 
words gave birth to formal logic. 

4. In every syllogism we find six such letters, called 'terms' 
(opot., 'boundaries'), equiform in pairs. Aristotle uses the following 
terminology: the term which is predicate in the conclusion and the 
term equiform to it in one of the premisses, are called 'major', 
evidently because in the first figure - but there only - this has the 
greatest extension. The term which is subject in the conclusion, and 
the term equiform to it in one of the premisses, he calls a 'minor' or 
'last' term (eXocttov, Ict^octov) for the same reason. Finally the two 
equiform terms that occur only in the premisses are called 'middle' 
terms. By contrast the two others are called 'extremes' (axpa). 

69 



THE GREEK VARIETY OF LOGIC 

Admittedly the terms are not so defined in the text of Aristotle : he 
gives complicated definitions based on the meaning of the terms ; but 
his customary syllogistic practice keeps to the foregoing definitions. 
- Sometimes the premiss containing the major term is called 'the 
major', the other 'the minor'. 

5. The letters (variables) can, in the system, only be substituted by 
universal terms; they are term-variables for universal terms. But it 
would not be right to call them class-variables, for that would be to 
ascribe to Aristotle a distinction between intension and extension 
which is out of place. One may ask why the founder of logic, whose 
philosophical development proceeded steadily away from Platonism 
towards a recognition of the importance of the individual, completely 
omitted singular terms in what (by contrast to the Hermeneia) is 
his most mature work. The reason probably lies in his assumption 
that such terms are not suitable as predicates (13.19), whereas 
syllogistic technique requires every extreme term to occur at least 
once as predicate. 

6. It is usually said that a further limitation is required, namely 
that void terms must not be substituted for the variables. But this is 
only true in the context of certain interpretations of the syllogistic; 
on other interpretations this limitation is not required. 

To sum up : we have in the syllogistic a formal system of term-logic, 
with variables, limited to universal terms, and consisting of propo- 
sitions, not rules. 

This system is also axiomatized. Hence we have here together 
three of the greatest discoveries of our science: formal treatment, 
variables, and axiomatization. What makes this last achievement 
the more remarkable is the fact that the system almost achieves 
completeness (there is lacking only an exact elaboration of the 
moods of the fourth figure). This is something rare for the first 
formulation of a quite original logical discovery. 

D. THE FIGURES AND FURTHER SYLLOGISMS 

The syllogisms are divided into three classes (ax^aTa), 'figures' as 
this was later translated, according to the position of the middle term. 
According to Aristotle there are only three such figures : 

13.20 So we must take something midway between the 
two, which will connect the predications, if we are to have a 
syllogism relating this to that. If then we must take something 
common in relation to both, and this is possible in three ways 
(either by predicating A of C, and C of B, or C of both, or 
both of C), and these are the figures of which we have spoken, 
it is clear that every syllogism must be made in one or other of 
these figures. 

70 



ARISTOTLE 

But evidently this is so far from being the case that Aristotle 
himself was well aware of a fourth figure. He treats its syllogisms 
as arguments obtainable from those already gained in the first 
figure: 

13.21 It is evident also that in all the figures, whenever a 
proper syllogism does not result, if both the terms are affir- 
mative or negative nothing necessary follows at all, but if one 
is affirmative, the other negative, and if the negative is stated 
universally, a syllogism always results relating the minor to 
the major term, e.g. if A belongs to all or some B, and B 
belongs to no C: for if the premisses are converted it is 
necessary that C does not belong to some A. Similarly also in 
the other figures. 

The case under consideration is this: the premisses are (1) l A 
belongs to all B\ (2) 'B belongs to no C\ They are both converted and 
their order is reversed (an operation that is irrelevant for Aristotle), 
so that we get: l C belongs to no B' and 'B belongs to some A\ But 
those are the premisses of the fourth mood of the first figure (Ferio, 
13.06), which has as conclusion 'to some A C does not belong'. Now 
if 'major term' and 'minor term' are defined as has been done above 
(§ 13, C 4) in accordance with the practice of Albalag and the moderns, 
then evidently C is the major term, A the minor, and so (2) is the 
major premiss, (1) the minor premiss. From that it follows that the 
middle term is predicate in the major premiss, subject in the minor, 
just the reverse of the situation in the first figure. We have here 
therefore a fourth figure. That Aristotle refuses to recognize any 
such, is due to his not giving a theoretical definition of the terms 
according to their place in the conclusion, but according to their 
extension, and so not a formal definition but one dependent on their 
meaning. - The syllogism just investigated later came to be called 
Fresison. 

Aristotle explicitly stated two syllogism of this figure, Fresison 
already cited, and in the same passage (13.31) Fesapo; he hints only 
at three more (13.32): Dimaris, Bamalip and Camenes. 

The same text (13.22) would permit us to gain still other syllo- 
gisms from two of the second figure Cesare: 13.08 and Cameslres: 
13.09) and from three of the third {Darapti: 13.13, Disamis: 13.15 
and Dalisi: 13.16). It is worth remarking that these hints do not 
occur in the text of the proper presentation of the syllogistic; 
apparently Aristotle only made these discoveries when his system 
was already in being. 

Consequently the following passage seems to be a later addition. 
It is at the origin of what later came to be called the 'subalternate 
syllogisms', Barbari, Celaront, Cesaro, Camestrop and Calemop. 

71 



THE GREEK VARIETY OF LOGIC 

13.23 It is possible to give another reason concerning 
those (syllogisms) which are universal. For all the things that 
are subordinate to the middle term or to the conclusion may 
be proved by the same syllogism, if the former are placed in 
the middle, the latter in the conclusion; e.g. if the conclusion 
AB is proved through C, whatever is subordinate to B or C 
must accept the predicate A : for if D is included in B as in a 
whole, and B is included in A, then D will be included in A. 
Again if E is included in C as in a whole, and C is included in 
A, then E will be included in A. Similarly if the syllogism is 
negative. In the second figure it will be possible to infer only 
that which is subordinate to the conclusion, e.g. if A belongs 
to no B and to all C ; we conclude that B belongs to no C. If 
then D is subordinate to C, clearly B does not belong to 
it. But that B does not belong to what is subordinate to A, is 
not clear by means of the syllogism. 

These syllogisms, however, are not developed. 

If we want to summarize the content of the texts we have adduced, 
we see that Aristotle in fact expressly formulated the conditions 
required for a system of twenty-four syllogistic moods, six in each 
figure. Of these twenty-four he only developed nineteen himself, 
only fourteen of them thoroughly. The remaining ten fall into three 
classes: (1) exactly formulated (Fesapo, Fresison: 13.21): (2) not 
formulated, but clearly indicated (Dimaris, Bamalip, Camenes: 
13.22); (3) only indirectly indicated: the five 'subalternate' moods 
(13.23). 

That explains how historically sometimes fourteen, sometimes 
nineteen and again at other times twenty-four moods are spoken of. 
The last figure is obviously the only correct one. For evidently no 
systematic principle can be derived from the fact that the author of 
the syllogistic did not precisely develop certain moods. 



§ 14. AXIOMATIZATION OF THE SYLLOGISTIC 
FURTHER LAWS 

A. AXIOMATIC THEORY OF THE SYSTEM 

Aristotle axiomatized the syllogistic, and in more than one way. In 
this connection we shall first cite the most important passages in 
which he presents his theory of the system as axiomatized, and then 
give the axiomatization itself. For this theory is the first of its kind 
known to us, and notwithstanding its weaknesses, must be consi- 
dered as yet another quite original contribution made to logic by 

72 



ARISTOTLE 

Aristotle. The point is a methodological one of course, not a matter of 

formal logic, and that Aristotle was himself aware of: 

14.01 We now state by what means, when, and how every 
syllogism is produced; subsequently we must speak of demon- 
stration. Syllogism should be discussed before demonstration, 
because syllogism is the more general: demonstration is a 
sort of syllogism, but not every syllogism is a demonstration. 

Aristotle's doctrine of demonstration is precisely his methodology. 
But as the methodology of deduction is closely connected with formal 
logic, we must go into at least a few details. 

14.02 We suppose ourselves to possess unqualified scientific 
knowledge of a thing, as opposed to knowing it in the accidental 
way in which the sophist knows, when we think that we know 
the cause on which the fact depends, as the cause of that fact 
and of no other, and, further, that the fact could not be other 
than it is . . . There may be another manner of knowing as 
well - that will be discussed later. What I now assert is that 
at all events we do know by demonstration. By demonstration 
I mean a syllogism productive of scientific knowledge, a 
syllogism, that is, the grasp of which is eo ipso such knowledge. 
Assuming then that my thesis as to the nature of scientific 
knowing is correct, the premisses of demonstrated knowledge 
must be true, primary, immediate, better known than and 
prior to the conclusion, which is further related to them as 
effect to cause. Unless these conditions are satisfied, the basic 
truths will not be 'appropriate' to the conclusion. Syllogism 
there may indeed be without these conditions, but such 
syllogism, not being productive of scientific knowledge, will 
not be demonstration. 

14.03 There are three elements in demonstration : (1) 
what is proved, the conclusion - an attribute inhering essen- 
tially in a genus; (2) the axioms, i.e. the starting points of 
proof; (3) the subject-genus whose attributes, i.e. essential 
properties, are revealed by the demonstration. 

It emerges clearly from this text that for Aristotle a demonstration 
(1) is a syllogism, (2) with specially constructed premisses, and (3) 
with a conclusion in which a property (11.081 is predicated of a genus. 
That, however, can only be achieved by means of a syllogism in the 
first figure : 

73 



THE GREEK VARIETY OF LOGIC 

14.04 Of all figures the most scientific is the first. Thus, it 
is the vehicle of the demonstrations of all the mathematical 
sciences, such as arithmetic, geometry, and optics, and practi- 
cally of all sciences that investigate causes ... a second proof 
that this figure is the most scientific; for grasp of a reasoned 
conclusion is the primary condition of knowledge. Thirdly, 
the first is the only figure which enables us to pursue know- 
ledge of the essence of a thing. . . . Finally, the first figure has 
no need of the others, while it is by means of the first that the 
other two figures are developed, and have their intervals 
close-packed until immediate premisses are reached. Clearly, 
therefore, the first figure is the primary condition of knowledge. 

This doctrine is only of historical importance, though that is 
considerable: on the other hand the essential of Aristotle's views on 
the structure of an axiomatic system has remained a part of every 
methodology of deduction right to our own day: 

14.05 Our own doctrine is that not all knowledge is demon- 
strative: on the contrary, knowledge of the immediate 
premisses is independent of demonstration. (The necessity of 
this is obvious; for since we must know the prior premisses 
from which the demonstration is drawn, and since the regress 
must end in immediate truths, those truths must be indemon- 
strable.) . . Now demonstration must be based on premisses 
prior to and better known than the conclusion ; and the same 
things cannot simultaneously be both prior and posterior to 
one another: so circular demonstration is clearly not possible 
in the unqualified sense of 'demonstration', but only possible 
if 'demonstration' be extended to include that other method 
of argument which rests on a distinction between truths 
prior to us and truths without qualification prior, i.e. the 
method by which induction produces knowledge. . . . The 
advocates of circular demonstration are not only faced with 
the difficulty we have just stated: in addition their theory 
reduces to the mere statement that if a thing exists, then it 
does exist - an easy way of proving anything. That this is 
so can be clearly shown by taking three terms, for to constitute 
the circle it makes no difference whether many terms or few 
or even only two are taken. Thus by direct proof, if A is, B 
must be ; if B is, C must be ; therefore if A is, C must be. Since 
then - by the circular proof - if A is, B must be, and if B is, A 
must be, A may be substituted for C above. Then 'if B is, A 

74 



ARISTOTLE 

must be' = 'if B is, C must be', which above gave the con- 
clusion 'if A is, C must be', but C and A have been identified. 

This is, be it said at once, far the clearest passage about our 
problem, which evidently faced Aristotle with enormous difficulties. 
Two elements are to be distinguished : on the one hand it is a matter 
of epislemological doctrine, according to which all scientific knowledge 
must finally be reduced to evident and necessary premisses; on the 
other hand is a logical theory of deduction, which states that one 
cannot demonstrate all sentences in a system, but must leave off 
somewhere; for neither a processus in infinitum nor a circular 
demonstration is possible. In other words: there mu-t be axioms in 
every system. 

B. SYSTEMS OF SYLLOGISTIC 

This doctrine was now applied by Aristotle to formal logic itself, 
i.e. to the syllogistic ; yes, the syllogistic is the first known axiomatic 
system, or more precisely the first class of such systems: for Ari- 
stotle axiomatized it in several ways. One can distinguish in his 
work the following systems: 1) with the four syllogisms of the first 
figure (together with other laws) as axioms, 2) with the first two 
syllogisms of the same figure, 3) with syllogisms of any figure as 
axioms, in which among other features the syllogisms of the first 
figure are reduced to those of the second and third. These three 
systems are presented in an object-language; there is further to be 
found in Aristotle a sketch for the axiomatization of the syllogistic 
in a metalanguage. 

We take first the second system, the first having been fully 
presented above in § 13. 

14.06 It is possible also to reduce all syllogisms to the 
universal syllogisms in the first figure. Those in the second 
figure are clearly made perfect by these, though not all in the 
same way; the universal syllogisms are made perfect by con- 
verting the negative premiss, each of the particular syllogisms 
by reductio ad impossibile. In the first figure particular 
syllogisms are indeed made perfect by themselves, but it is 
possible also to prove them by means of the second figure, 
reducing them ad impossibile, e.g. if A belongs to all B, and 
B to some C, it follows that A belongs to some C. For if it 
belonged to no 67, and belongs to all B, then B will belong 
to no C: this we know by means of the second figure. 

It is here shewn that Darii (13.05) can be reduced to Cameslres 
(13.09) ; the proof of Ferio (13.06) is similarly effected, and it is then 

75 



THE GREEK VARIETY OF LOGIC 

shewn that the syllogisms of the third figure can also be easily 
deduced. 

In these operations the syllogisms of the first figure always play 
the part of axioms, for the reason that they are to be 'perfect' 
(TeAeLOi) syllogisms (14.08). This expression is explained thus: 

14.09 I call that a perfect syllogism which needs nothing 
other than what has been stated to make plain what neces- 
sarily follows; a syllogism is imperfect, if it needs either one 
or more propositions, which are indeed the necessary conse- 
quences of the terms set down, but have not been expressly 
stated as premisses. 

But this can only mean that perfect syllogisms are intuitively 
evident. 

C. DIRECT PROOF 

To be able to deduce his syllogisms Aristotle makes use of three 
procedures, and in each of another class of formulae, not named as 
axioms and in part tacitly presented. The procedures are direct 
proof (SzixTitic, avaysLv), reduction to the impossible (zlc, to aSuvarov 
avayetv) and ecthesis (setting out of terms, exOsctic;). 

In direct proof the laws of conversion of sentences are explicitly 
presupposed ; they are three : 

14.10 If A belongs to no B, neither will B belong to any A. 
For if to some, say to C, it will not be true that A belongs to 
no B; for C is one of the things (which are) B. 

14.11 If A belongs to all B, B also will belong to some A; 
for if to none, then neither will A belong to any B; but by 
hypothesis it belonged to all. 

14.12 If A belongs to some B, B also must belong to some 
A; for if to none, then neither will A belong to any B. 

Aristotle prefaces the syllogistic proper with these laws and their 
justification, clearly conscious that he needs them for the 'direct 
procedure' .They are the laws of conversion of affirmative (universal 
and particular), and universal negative propositions. (The conversion 
of particular negatives is expressly recognized as invalid: 14.13.) 

It is noteworthy that Aristotle tries to axiomatize these laws too : 
the first is proved by ecthesis and serves as axiom for the two 
others. 

Besides these explicit presuppositions of the syllogistic, some 
rules of inference are also used, without Aristotle having consciously 
reflected on them. They are these : 

14.141 Should 'If p and q, then r' and 'If s, then p' be 
valid, then also 'If s and q, then r' is valid. 

76 



ARISTOTLE 

14.142 Should 'If p and q, then r' and 'If s, then q f be 
valid, then also 'If p and s, then r' is valid. 

14.151 Should 'If p and q, then r' be valid, then also 'If q 
and p, then r' is valid. 

14.161 Should 'If p, then </' and 'If </, then r' be valid, then 
also 'If p, then r' is valid. 

Some of these rules also used - without being explicitly appealed 
to - for constructing formulae that later came to be called 'polysyllo- 
gisms, or 'soriteses' : 

14.17 It is clear too that every demonstration will proceed 
through three terms and no more, unless the same con- 
clusion is established by different pairs of propositions; . . . 
Or again when each of the propositions A and B is obtained 
by syllogistic inference, e.g. A by means of D and E, and 
again B by means of F and G. . . . But thus also the syllogisms 
are many; for the conclusions are many, e.g. A and B and C. 

Also to be noted in this text is Aristotle's evident use of proposi- 
tional variables. 

D. INDIRECT PROOF 

Aristotle has two different procedures for reduction to the impos- 
sible, the first being invalid and clearly earlier. In both, the laws of 
opposition are presupposed. By contrast to the laws of conversion 
these are neither systematically introduced nor axiomatized; they 
occur as the occassion of the deduction requires. The reason for 
their not being systematized or axiomatized may be that the essential 
points about them have been stated already in the Hermeneia. The 
main features have been summarized above (12.10, 12,11). 

The procedures are as follows: 

First procedure 

It is used to reduce Baroco (13.11) and Bocardo (13.17), and takes 
the course outlined above (in the commentary on 13.11). As Luka- 
siewicz (14.18) has shown, it is not conclusive. This can be made 
evident by the following substitution: we put 'bird' for 'M', 'beast' 
for W and 'owl' for l X' in Baroco (13.11). We obtain: 
If (1) bird belongs to all beast 
and (2) to some owl bird does not belong, 
then (3) to some owl beast does not belong. 
The syllogism is correct, being a substitution in Baroco; but all its 
three component sentences are manifestly false. Now if we apply the 
procedure described above (commentary on 13.11), we must form 
the contradictory opposite to (3) : 
(4) Beast belongs to all owl. 

77 



THE GREEK VARIETY OF LOGIC 

This produces with (1) a syllogism in Barbara (13.02), having as 
conclusion 

(5) Bird belongs to all owl, which so far from being false, is 
evidently true. Hence the procedure fails to give the required 
conclusion and must be deemed incorrect. 

It would certainly be correct if Aristotle had not expressed the 
syllogism as a conditional sentence (in which the antecedent does not 
need to be asserted), but in the scholastic manner as a rule (31.11) 
in which one starts from asserted premisses. 

Second procedure 

We do not know whether Aristotle saw the incorrectness of the 
first procedure; in any case in book B of the Prior Analytics he 
several times uses another which is logically correct. 

It is to be found in the place where he treats of the so-called 
'conversion' (avTLaTpocp-y)) of syllogisms, a matter of replacing one 
premiss by the (contradictory) opposite of the conclusion. 

14.19 Suppose that A belongs to no B, and to some C: 
the conclusion is BC. ... If the conclusion is converted into 
its contradictory, both premisses can be refuted. For if B 
belongs to all C, and A to no B, A will belong to no C: but 
it was assumed to belong to some C. 

The following scheme reproduces the thought: 

Original syllogism Converted syllogism 

(Festino) (Celarent) 

If A belongs to no B If A belongs to no B 

and A belongs to some C, and B belongs to all C, 

then to some C B does not then A belongs to no C 

belong. 

The rule presupposed, of which Aristotle was conscious (cf. 
16.33) - he often used it, - is this: 

14.201 Should 'If p and q, then r' be valid then also 'If not-r 
and q, then not-p' is valid. 

A similar rule, also often used, is: 

14.202 Should 'If p and q, then r' be valid, then also 'If 
p and not-r, then not-qr' is valid. 

By the use of these rules with the laws of opposition and some of 
the rules given above (14.151, 14.161), any syllogism can in fact be 
reduced to another. 

78 



ARISTOTLE 

Applications 

By varying this second procedure Aristotle was able to construct 
a third axiomatization - or rather a class of further axiomatiza- 
tions - of his syllogistic: Syllogisms of either the first, second or 
third figure are taken, and the others proved from them by reduc- 
tion to the impossible. The result is summarized thus: 

14.21 It is clear that in the first figure the syllogisms are 
formed through the middle and the last figures ... in the 
second through the first and the last figures ... in the third 
through the first and the middle figures. 

We shall not go into the practical details (14.22), but only note 
that Aristotle replaces premisses not only by their contradictory, 
but also by their contrary opposites, and that he investigates all 
syllogisms. 

The results of replacing premisses by the contradictory opposite 
of the conclusion can be clearly presented in the following way: 
From a syllogism of figure 1 2 3 

there results by substitution 
of the negation of the conclusion 
a syllogism of figure: 

substitution for the major premiss 3 3 1 

substitution for the minor premiss 2 12 

E. DICTUM DE OMNI ET NULLO 

A word must now be said about the 'dictum de omni el nullo' that 
later became so famous. It concerns the following sentence: 

14.23 That one thing should be in the whole of another and 
should be predicated of all of another is the same. We say 
that there is predication of all when it is impossible to take 
anything of which the other will not be predicated; and 
similarly predication of none. 

It is not clear whether Aristotle really intended here to establish 
an axiom for his system, as has often been supposed. One is rather 
led to suppose that he is simply describing the first and second 
moods of the first figure (13.02, 13.03). However, the dictum can 
be understood as an axiom if it is considered as a summary of the 
first four moods of the first figure, which is not in itself impossible. 

In this connection we quote a historically and systematically 
more important passage in which Aristotle deals with a problem of 
the theory of the three figures (13.20). In it he makes an essential 
advance in analyzing propositions and gives expression to thoughts 
that are not without significance for the theory of quantification. 

79 



THE GREEK VARIETY OF LOGIC 

14.24 It is not the same, either in fact or in speech, that A 
belongs to all of that to which B belongs, and that A belongs 
to all of that to all of which B belongs : for nothing prevents B 
from belonging to C, though not to all C: e.g. let B stand 
for 'beautiful' and C for 'white'. If beauty belongs to something 
white, it is true to say that beauty belongs to that which is 
white: but not perhaps to everything that is white. 

Here an analysis of the sentence A belongs to all B 'is presented, 
which could be interpreted in this way: 'For all x: if B belongs to x, 
then A belongs to x; it would then be a matter of the modern formal 
implication. That Aristotle thought of such an analysis - at least 
during his later period - seems guaranteed by the fact that he 
explicitly applied it to modal logic (cf. 15.13). The Scholastics, as we 
shall see, treated these thoughts as an elucidation of the dictum. 

F. BEGINNINGS OF A METALOGICAL SYSTEM 

Aristotle also described his syllogisms metalogically in such a way 
that a new, metalogical system could easily be established: 

14.25 In every syllogism one of the premisses must be 
affirmative, an universality must be present. 

14.26 It is clear that every demonstration will proceed 
through three terms and no more, unless the same conclusion 
is established by different pairs of propositions. 

14.27 This being evident, it is clear that a syllogistic 
conclusion follows from two premisses and not from more 
than two. 

14.28 And it is clear also that in every syllogism either 
both or one of the premisses must be like the conclusion. I 
mean not only in being affirmative or negative, but also in 
being necessary, assertoric or contingent. 

Aristotle does not carry out this application to modal logic; 
possibly this is an interpolation by another hand. 

In developing the several figures Aristotle established similar 
rules for each. 13.05 contains an example. Taken all together these 
rules form an almost complete metalogical description of the syllo- 
gistic, which one would like to develop. 

G. THE INVENTIO MEDII 

Here we want to allude briefly to a doctrine of the Prior Analytics 
which is not essentially a matter of formal logic but rather of 
methodology, and that is the discussions about what was later 

80 



ARISTOTLE 

called the inventio medii. In connection with axiomatization one 

can ask two different b;isic questions: (1) What follows given 
premisses? (2) From what premisses can ;i given sentence (con- 
clusion) be deduced? Aristotle primarily considered the first ques- 
tion, but in the following text and its continuation he poses also 
the second, and tries to show the premisses of a syllogism must be 
constructed in order to yield a given conclusion. At the same time 
he gives practical advice on the forming of syllogisms: 

14.29 The manner in which every syllogism is produced, 
the number of the terms and premisses through which il 
proceeds, the relation of the premisses to one another, the 
character of the problem proved in each figure, and the num- 
ber of the figures appropriate to each problem, all these 
matters are clear from what has been said. We must now 
state how we may ourselves always have a supply of syllogisms 
in reference to the problem proposed and by what road we may 
reach the principles relative to the problem: for perhaps we 
ought not only to investigate the construction of syllogisms, 
but also to have the power of making them. 

We do not need to pursue the details of this theory here. It only 
interests us as the starting-point of the scholastic pons asinorum. 



§ 15. MODAL LOGIC* 

A. MODALITIES 

Aristotle distinguishes three principal classes of premisses: 

15.01 Every premiss is either about belonging to, or 
necessarily belonging to, or possibly belonging to. 

The expressions 'necessary' (el* avayxY)*;) and 'possible' (can 
belong to, IvSe^ETai, SuvaTai) have several meanings. 

1. In respect of the functor 'necessary' (or 'must') we have already 
remarked (§ 13, C, 2) that it often only expresses logical consequence. 



* The Aristotelian (as also the Theophrastan) modal logic is here interpreted in 
the way that was customary among- the Scholastics, rediscovered by A. Becker 
in 1934, and served as a basis for my ideas in works on the history of modal 
logic, on Theophrastus and on ancient logic. 

However, while I was writing this work, Prof. J. Lukasiewicz communicated 
his being in possession of a quite different interpretation, showing the Aristote- 
lian system to have contained mistakes which were rectified by Theophrastus. 
This new interpretation has now been published [Aristotle's Syllogistic, 2nd ed., 
Oxford, 1957). 

81 



THE GREEK VARIETY OF LOGIC 

That this is so, can be clearly seen where for instance Aristotle 
says 'It is necessary that A necessarily belongs to B' (15.02), or 
again 'It is necessary that A possibly belongs to B' (15.03). The 
first 'necessary' evidently means logical (a.n'k&c,) and hypothetical 
(toutcov ovtcov) necessity (15.04). The necessity that something is 
when it is (6t<xv JJ) obviously belongs to the second of these classes 
(15.05). 

2. Even the simple, unqualified (assertoric) 'belonging to', which 
Aristotle often calls 'mere belonging to', is divided into an absolute 
(a.Tzk&Q) and temporally qualified (xorra ^povov) kind, with different 
logical properties (15.06). 

3. As to possibility, Aristotle distinguishes at first two kinds: 
the one-sided and the two-sided. 

This distinction emerges from a searching discussion which 
Aristotle conducts in the Hermeneia. The passage is of great impor- 
tance for the understanding of the whole doctrine of modalities, 
and so we give it in full : 

15.07 When it is necessary that a thing should be, it is 
possible that it should be. (For if not, the opposite follows, 
since one or the other must follow; so, if it is not possible, it 
is impossible, and it is thus impossible that a thing should be 
which must necessarily be; which is absurd.) Yet from the 
proposition 'it may be' it follows that it is not impossible, and 
from that it follows that it is not necessary; it comes about 
therefore that the thing which must necessarily be need not 
be; which is absurd. But again, the proposition 'it is necessary 
that it should be' does not follow from the proposition 'it 
may be', nor does the proposition 'it is necessary that it 
should not be'. . . . For if a thing may be, it may also not be, 
but if it is necessary that it should be or that it should not be, 
one of the two alternatives will be excluded. It remains, 
therefore, that the proposition 'it is not necessary that it 
should not be' follows from the proposition 'it may be'. For 
this is true also of that which must necessarily be. 

The sequence of thought here is, in summary form: If something 
is necessary, then it is also possible; but what is possible, can also 
not be (it is not impossible that it should not be) ; but from that it 
follows that it is not necessary, and so a contradiction results. The 
solution consists in distinguishing the two meanings of being 
'possible': 

15.071 One-sided possibility: that is possible which not 
necessarily is not (which is not impossible). 

82 



ARISTOTLE 

15.072 Two-sided possibility: that is possible which neither 
necessarily is nor necessarily is not (nor impossibly is). 

This second, two-sided possibility is the one intended in the 
syllogistic, and Aristotle only uses the first kind when forced to it. 
He defines two-sided possibility in the Prior Analytics thus: 

15.08 I use the terms 'to be possible' and 'the possible' of 
that which is not necessary but, being assumed, results in 
nothing impossible. 

It coincides, as can be seen, with the definition just fdven above; 
we find a similar one in the Metaphysics (15.09). 

In two texts - but both extremely unclear and so hard to reconcile 
with the teaching as a whole as to constitute an unsolved problem 
(15.10) - Aristotle subdivides two-sided possibility. In the first 
passage he speaks of a possibility in the sense of 'in most cases' 
(&$ £7il to 7U>Xu) and of another besides (15.11); in the second 
passage he distinguishes between a 'natural' (to TC(pi)XO£ \jtA^/zvj 
and an indeterminate (to aopiaTOv) or 'contingent' (to arco t's/y^ 
possibility which, so he says, is no concern of science. Both passages 
are probably interpolations. 

B. STRUCTURE OF MODAL SENTENCES 

The normal use of 'possible' in the sense of two-sided possibility 
is a distinguishing characteristic of Aristotle's modal logic. Another, 
of no less importance, is his view of the structure of modal sentences. 
He only gives explicit expression to this view in one place, but it 
lies at the base of the whole modal syllogistic and exercises a most 
remarkable influence. 

15.13 The expression 'it is possible for this to belong to 
that' may be understood in two senses: either as 'to the thing 
to which that belongs' or as 'to the thing to which that can 
belong' ; for 'to that of which B (is predicated) A can (belong)' 
means one of the two: 'to that of which B is predicated' or 
'to that of which (B as) possibly (belonging) is predicated'. 

This contains two points: 

First a sentence of the form 'A belongs to B' is paraphrased by the 
formula 'to that to which B belongs (of which B is predicated) A 
also belongs' : implying a very subtle analysis of the sentence, 
reminiscent of the modern formal implication, which we find else- 
where in the Analytics (cf. 14.24). 

Secondly it can be gathered from this text that the modal functor 
does not determine the sentence as a whole, but part of it. So that 

83 



THE GREEK VARIETY OF LOGIC 

for Aristotle a modal sentence is not to be conceived in such a sense as : 
'It is possible that: A belongs to B\ The modal functor does not 
precede the whole sentence but one of its arguments. This distinction 
quickly becomes still clearer, for the distinction is three times made 
between two possible cases: 

1. to that to which B belongs, A also can belong, 

2. to that to which B can belong, A also can belong. In the 
first case the modal functor determines only the consequent, in the 
second case it determines the antecedent too. 

This analysis is not expressly extended to necessity, but that 
extension must be supposed : for otherwise many syllogisms would be 
invalid. 

C. NEGATION AND CONVERSION 

In the Hermeneia Aristotle establishes a 'logical' square for senten- 
ces with modal functors, in which the two expressions for 'possible' 
(Suvoctov and evSe^ofjisvov) mean one-sided possibility. This square 
can be compressed into the following scheme, in which all expressions 
in any one row are equivalent: 

possible not impossible not necessary not 

not possible impossible necessary not 

possible not not impossible not not necessary 

not possible not impossible not necessary (15.14) 

More complicated is the doctrine of the negation of sentences 
containing the functor of two-sided possibility. Since this has been 
defined by a conjunction of two sentences, Aristotle rightly deduces, 
on the basis of the so-called de Morgan law (not to be found in 
him) : 

15.15 If anyone then should claim that because it is not 
possible for C to belong to all D, it necessarily does not belong 
to some D, he would make a false assumption: for it does 
belong to all D, but because in some cases it belongs neces- 
sarily, therefore we say that it is not possible for it to belong 
to all. Hence both the propositions l A necessarily belongs to 
some B' and 'A necessarily does not belong to some B : are 
opposed to the proposition l A may belong to all B\ 

The passage is not quite clear; but the author's intention can 
be formulated : 

15.151 p is not possible, if and only if, one of the two, p and 
not-p, is necessary. 

From this it results that the negation of such a sentence issues as 
an alternation, such as is in no case permissible as a premiss in an 
Aristotelian syllogism. This prevents Aristotle from using reduction 
to the impossible in certain cases. 

84 



ARISTOTLE 

Another result which Aristotle subtly deduces from his supposi- 
tions is his doctrine of the equivalence of affirmative and negative 
sentences when they contain the functor under consideration : 

15.16 It results that all premisses in the mode of possibility 
are convertible into one another. I mean not that the affirma- 
tive are convertible into the negative, but that those which 
are affirmative in form admit of conversion by opposition, 
e.g. 'it is possible to belong' may be converted into 'it is 
possible not to belong'; 'it is possible to belong to all' into 'it 
is possible to belong to none' and 'not to all'; 'it is possible 
to belong to some' into 'to some it is possible not to belong'. 
And similarly in other cases. 

Take the three modal sentences: 

(a) 'A possibly belongs to B\ 

(b) 'A does not possibly belong to B\ 

(c) l A possibly does not belong to B\ 

(b) is the proper denial of (a), (c) is no denial of (a) but it is 
'negative in form'. Then it is stated that sentences such as (a) 
imply those such as (c), and are even equivalent to them. So we 
have following laws : 

15.161 p is possible if and only if p is not possible. 

15.162 It is possible that A belongs to all B, if and only if, 
it is possible that A belongs to no B. 

15.163 It is possible that A belongs to some B, if and only 
if, it is possible that to some B A does not belong. 

Law^ analogous to those for ordinary conversion (14.10) hold 
for sentences containing the functors of necessity and one-sided 
possibility (15.17), just parallel to the corresponding laws in asser- 
toric logic (15.18). 

By contrast, the laws of conversion for sentences with the two- 
sided functor are different: the universal negative cannot be con- 
verted (15.19), but the particular negative can (15.20). The affir- 
mative sentences are converted like assertoric ones (15.21). 

D. SYLLOGISMS 

On this basis and with the aid of the same procedures developed 
for the assertoric syllogistic, Aristotle now builds the vast structure 
of his system of syllogisms with modal premisses. Vast it is even 
in the number of formulae explicitly considered, they are not fewer 
than one hundred and thirty seven. But it appears much vaster - in 
spite of many points where it is incomplete - in view of the subtlety 

85 



THE GREEK VARIETY OF LOGIC 

with which the original master-logician operates in so difficult a 
field. De modalibus non gustabit asinus was a medieval proverb; 
but one does not need to be a donkey to get lost in this maze of 
abstract laws: Theophrastus quite misunderstood the system, and 
nearly all the moderns, until 1934. 

The syllogisms which it comprises can be arranged in eight 
groups. If we write 'A 7 ' for a premiss with the functor 'necessary', 
'Af for one with the functor 'possible' and 'A' for an assertoric 
premiss (that is, one which predicates mere belonging to), these 
groups can be shown as follows: 

Group 12 3 4 5 6 7 8 

Major premiss N N A M M A M N 

Minor premiss N A N M A M N M 

An.Pr.A, chap. 8 9-11 9-11 14/17/20 15/18/21 16/19/22 

A striking characteristic of this syllogistic is that in very many 
syllogisms the conclusion (contrary to 14.28) has a stronger 
modality than the premisses, necessity being reckoned as stronger 
that mere belonging to and this as stronger than possibility. 

15.22 It happens sometimes also that when one premiss is 
necessary the conclusion is necessary, not however, w T hen 
either premiss is necessary, but only when the major is, e.g. 
if A is taken as necessarily belonging or not belonging to B, 
but B is taken as simply belonging to C: for if the premisses 
are taken in this way, A will necessarily belong or not belong 
to C. For since A necessarily belongs, or does not belong, to 
every B, and since C is one of the J5s, it is clear that for C 
also the positive or the negative relation to A will hold 
necessarily. 

And of course that is the case, if one presupposes the structure 
of the modal sentences as given above. For then the syllogism here 
described (an analogue of Barbara) will be interpreted as follows: 
If to all to which B belongs, A necessarily belongs, 
and to that to which C belongs, B belongs, 
then to all to which C belongs, A necessarily belongs, 
which is clearly correct. 

Hence it is quite wrong to extend the validity of the principle 
'the conclusion follows the weaker premiss' (cf. 14.28 and 17.17) 
to Aristotle's modal syllogistic. 

Another striking fact is that there are numerous valid modal 
syllogisms whose analogues in the assertoric syllogistic are invalid, 
as for instance those two negative premisses (in opposition to 
14.25) ; this is especially the case when the modal syllogism has a 
premiss with the functor of possibility where the assertoric analogue 
has an affirmative. For, as has been said, affirmative and negative 

86 



ARISTOTLE 

possible premisses are equivalent and can replace one another. 
We take as an example a passage where Aristotle, after giving an 
analogue of Barbara, in the fourth group, to which he refers by 
such phrases as 'previously', 'the same syllogism', 'as before', then 

proceeds: 

15.23 Whenever A may belong to all B, and B may belong 

to no 67, then indeed no syllogism results from the premisses 
assumed, but if the premiss BC is converted after the manner 
of problematic propositions, the same syllogism results as 
before. For since it is possible that B should belong to no C, 
it is possible also that it should belong to all C. This has been 
stated above. Consequently if B is possible for all C, and A 
is possible for all B, the same syllogism again results. Similarly 
if in both the premisses the negative is joined with 'it is 
possible': e.g. if A may belong to none of the Bs, and B to 
none of the Cs. No syllogism results from the assumed premis- 
ses, but if they are converted we shall have the same syllogism 
as before. 

This syllogistic is, like the assertoric, axiomatized. There serve 
as axioms the syllogisms of the first figure in all groups, except the 
sixth and eighth, together with the laws of conversion and, when 
assertoric premisses occur, principles of the assertoric syllogistic. The 
other syllogisms are reduced to those axioms, mostly by conversion 
of premisses (direct procedure). Reduction to the impossible serves 
to prove syllogisms of the first figure in the eighth group and the 
analogue of Bocardo in the fifth. The analogues of Baroco and 
Bocardo in the first group are proved only by ecthesis, while the 
same analogues in the second and third groups remain unproved, 
though it should not be hard to prove them. 

The hardest problem for Aristotle are the syllogisms of the sixth 
group. The first figure ones among them rightly do not rank as 
intuitively evident; e.g. the analogue of Barbara would be: 
If to all to which B belongs A belongs, 
and to all to which C belongs, B may belong, 
then to all to which C belongs, A belongs. 
For this to be evident one would have to see the Tightness of the 
sentence 'To all to which B may belong, B belongs' ; but according 
to the definition of possibility, that is false. The details of Aristotle's 
complicated attempts to validate this syllogism are matter of 
conjecture and dispute, but the fact that he has to replace the 
problematic minor premiss with an assertoric one is a sufficient 
indication of its essential weakness (15.24). However, the passage 
which contains this 'proof is one of the few where Aristotle rises to 

87 



THE GREEK VARIETY OF LOGIC 

the use of propositional variables, and for that reason remains of the 
greatest logical interest (15.25). 

This abortive proof is moreover not the only inconsistency in the 
Aristotelian modal logic. There are for instance essential difficulties 
in connection with the conversion of premisses with the functor of 
necessity, and consequently in the proving of many syllogisms which 
contain such premisses. In general one gets the impression that this 
modal logic, by contrast to the assertoric syllogistic, is still only 
in a preliminary and incomplete stage of development. 



16. NON-ANALYTIC LAWS AND RULES 



For those reasons there is no possible doubt that the theory of 
what Aristotle would have called 'analytic' syllogisms is his chief ac- 
complishment in the field of formal logic. And so great an accomplish- 
ment is it from the historical and systematic points of view, that 
later, 'classical' logicians have mostly overlooked all else in his 
work. Yet the Organon contains a profusion of laws and rules of 
other kinds. Aristotle himself recognized some of them as autono- 
mous formulae, irreducible to his syllogistic. In other words: he 
saw that a 'reduction' of these laws and rules to the syllogistic is 
impossible - a thing which all too many after him did not see. 

From the historical standpoint these formulae are to be divided 
into three classes : first we have the formulae which are to be attri- 
buted to a period in which Aristotle had not yet discovered his 
analytic syllogisms. These are to be found in the Topics (and in the 
Rhetoric). Some of them were later re-edited with the help of vari- 
ables, and recognized as valid also in the period of the Analytics. 
Secondly there are the formulae which Aristotle indeed considered 
but mistakenly, as analytic, the syllogismi obliqui as they were later 
called. Finally, in reviewing the completed system of his syllogistic 
he discovered the 'hypothetical' procedure and in some cases 
attained to full consciousness of propositional formulae. 

But all these formulae are contained only in asides, and were 
never systematically developed as was the syllogistic. Furthermore, 
Aristotle thought, quite rightly in view of his methodological 
standpoint, that only the analytic formulae were genuinely 'scienti- 
fic', i.e. usable in demonstration. 

We give first the passages which, as it seems, introduce us to 
Aristotle's last thoughts on this question, then the actual non- 
analytic formulae divided into five classes: those belonging to the 
logic of classes, to the theory of identity, to the 'hypothetical' 
syllogistic, to the theory of relations, and to propositional logic. 

88 



ARISTOTLE 
A. TWO KINDS OF INFERENCE 

16.01 In some arguments it is easy to see what is wanting, 
but some escape us, and appear to be syllogisms, because 
something necessary results from what has been laid down, 
e.g. if the assumptions were made that substance is not 
annihilated by the annihilation of what is not substance, and 
that if the elements out of which a thing is made are annihi- 
lated, then that which is made out of them is destroyed : these 
propositions being laid down, it is necessary that any part 
of substance is substance; this has not, however, been drawn 
by syllogism from the propositions assumed, but premisses 
are wanting. Again if it is necessary that animal should 
exist, if man does, and that substance should exist if man 
does: but as yet the conclusion has not been drawn syllo- 
gistically: for the premisses are not in the shape we required. 
We are deceived in such cases because something necessary 
results from what is assumed, since the syllogism also is 
necessary. But that which is necessary is wider than the 
syllogism : for every syllogism is necessary, but not every- 
thing which is necessary is a syllogism. 

We must pass over the first example, about parts of substance, as 
its elucidation would take up too much space. But the second is 
clear; it concerns a law, not of propositional, but of predicate logic: 

16.011 If, when x is A then it is B, and when x is B then 
it is C, then, when x is A then it is C. 

This is a correct logical formula, and Aristotle is quite right in 
saying that it permits necessary inference. Hence he also realized 
that it falls under his definition of syllogism (10.05). But he refuses 
to admit it as syllogism. That means that his conception of syllo- 
gism had developed between the time when he penned the definition 
and that when he penned this passage. The definition applies to all 
correct logical formulae (and substitutions in them), but only a 
sub-class retains the name 'syllogism'. We know what this sub-class 
is that of the 'analytic' syllogisms. All other formulae may indeed be 
logically necessary, but are not genuine syllogisms. 

This distinction is not merely a matter of terminology. That 
becomes evident in the passages where Aristotle deals with the 
'hypothetical' syllogisms. 

16.02 We must not try to reduce hypothetical syllogisms; 
for with the given premisses it is not possible to reduce them. 
For they have not been proved by syllogism, but assented to 

89 



THE GREEK VARIETY OF LOGIC 

by agreement. For instance if a man should suppose that unless 
there is one faculty of contraries, there cannot be one science, 
and should then argue that there is no* (one) faculty of 
contraries, e.g. of what is healthy and what is sickly: for the 
same thing will then be at the same time healthy and sickly. 
He has shown that there is not one faculty of all contraries, 
but he has not proved that there is not a science. And yet 
one must agree. But the agreement does not come from a 
syllogism, but from an hypothesis. This argument cannot be 
reduced : but the proof that there is not a single faculty can. 

Evidently a substitution is there being made in the law: 
If (1) when not p then not q, and (2) not p, then (3) not q. (2) is 
proved by an analytic syllogism, but as (1) is merely supposed and 
not proved, the conclusion (3) also counts as not proved. That may 
be so, but Aristotle has not noticed that the assumed formula is no 
mere supposition but a correct logical law. The matter is still worse 
in the next text, an immediate continuation of the last: 

16.03 The same holds good of arguments which are brought 
to a conclusion per impossibile. These cannot be analyzed 
either; but the reduction to what is impossible can be analyzed 
since it is proved by syllogism, though the rest of the argument 
cannot, because the conclusion is reached from an hypothesis. 
But these differ from previous arguments : for in the former a 
preliminary agreement must be reached if one is to accept 
the conclusion; e.g. an agreement that if there is proved to 
be one faculty of contraries, then contraries fall under the 
same science; whereas in the latter, even if no preliminary 
agreement has been made, men still accept the reasoning, 
because the falsity is patent, e.g. the falsity of what follows 
from the assumption that the diagonal is commensurate, viz. 
that then odd numbers are equal to evens. 

So in reduction to the impossible too, Aristotle regards the 
inference as not 'demonstrated', though he has to recognize that no 
agreement needs to be presupposed to warrant inference. 

One could express this doctrine as follows: the class of correct 
formulae contains two sub-classes : that of the 'better' and that of the 
'less good' in relation to 'scientific demonstration'. The less good 

* Reading uia with the manuscript tradition A 2 B 2 C 2 T, against the (better) 
tradition ABCnAl, Waitz and Ross. For raccra, read by the latter, would mean 
a logical mistake in Aristotle which seems to me unlikely in this connection. For 
the critical apparatus vid. Ross ad loc. 

90 



ARISTOTLE 

are precisely our non-analytic formulae, for which we have chosen 
this name because according to Aristotle they are not reducible to 
the classical syllogistic, 'not analyzable into the figures'. (In this lie 
is evidently right, by contrast to a certain tradition.) That does not 
mean that these formulae are worthless for him; on the contrary 
he views them with a lively interest. 

16.04 These points will be made clearer by the sequel, 
when we discuss the reduction to impossibility. ... In the 
other hypothetical syllogisms, I mean those which proceed 
by substitution, or positing a certain quality, the inquiry will 
be directed to the terms of the problem to be proved - not the 
terms of the original problem, but the new terms introduced; 
and the method of the inquiry will be the same as before. 
But we must consider and determine in how many ways 
hypothetical syllogisms are possible. 

16.05 Many other arguments are brought to a conclusion 
by the help of an hypothesis; these we ought to consider and 
mark out clearly. We shall describe in the sequel differences, 
and the various ways in which hypothetical arguments are 
formed: but at present this much must be clear, that it is not 
possible to resolve such arguments into the figures. And we 
have explained the reason. 

On that Alexander of Aphrodisias remarks: 

16.06 He says that many others (syllogisms) besides are 
formed from hypotheses, and promises to treat thoroughly 
of them later. But no writing of his on this subject is extant. 



B. LAWS OF CLASS- AND PREDICATE-LOGIC 

16.07 If man (is) an animal, what is not-animal is not man. 

16.08 If the pleasant (is) good, the not-good (is) not pleasant. 

Notice that quantifiers are here lacking: so it is not a question of 
contraposition in the ordinary sense of the word. 

Aristotle was well aware that conversion of such sentences is 
invalid : 

16.09 For animal follows on man, but not-animal does not 
(follow) on not-man; the reverse is the case. 

Here there belong perhaps some rules which otherwise interpreted 
could be counted in with those of the 'logical square' : 

91 



THE GREEK VARIETY OF LOGIC 

16.10 When we have shown that a predicate belongs in 
every case, we shall also have shown that it belongs in some 
cases. Likewise, also, if we show that it does not belong in 
any case, we shall also have shown that it does not belong in 
every case. 

It is to be noted here that these formulae are not laws but, as is 
mostly the case in the Topics, rules. 

A series of similar laws is concerned with contrariety (in the sense 
of the earlier notion: 12.02): 

16.11 Health follows upon vigour, but disease does not 
follow upon debility; rather debility follows upon disease. 

16.12 Public opinion grants alike the claim that if all 
pleasure be good, then also all pain is evil, and the claim that 
if some pleasure be good, then also some pain is evil. More- 
over, if some- form of sensation be not a capacity, then also 
some form of failure of sensation is not a failure of capa- 
city. . . . Again, if what is unjust be in some cases good, then 
also what is just is in some cases evil; and if what happens 
justly is in some cases evil, then also what happens unjustly 
is in some cases good. 

It may be doubted whether Aristotle continued to recognize these 
laws as valid. But they are not without historical and even syste- 
matic interest. 

C. THEORY OF IDENTITY 

As already noted (11.11) Aristotle distinguishes three kinds of 
identity. Concerning the first, numerical identity, he developed 
the outline of a theory; its discovery is often falsely attributed to 
Leibniz. 

16.13 Again, look and see if, supposing the one to be the 
same as something, the other also is the same as it: for if 
they be not both the same as the same thing, clearly neither 
are they the same as one another. Moreover, examine them in 
the light of their accidents or of the things of which they are 
accidents: for any accident belonging to the one must belong 
also to the other, and if the one belongs to anything as an 
accident, so must the other also. If in any of these respects 
there is a discrepancy, clearly they are not the same. 

We have here in very compressed form a rather highly developed 
doctrine of identity; indeed this text contains a greater number of 

92 



ARISTOTLE 

fundamental laws of identity than the corresponding chapter of 
Principia Malhemalica (*13), and moreover Aristotle was the first 
to call to mind identity, in the passage mentioned above (\2AH t . The 
laws here sketched, can be formulated as follows with the help of 
variables: 

16.131 If B is identical with A, and C is not identical with 
A, then B and C are not identical. 

16.132 If A and B are identical, then (for all C): if Cbelongs 

to A, then it belongs also to B. 

16.133 If A and B are identical, then (for all C) : if A belongs 
to C, then it belongs also to B. 

16.134 If there is a C which belongs to A but not to B, then 
A and B are not identical. 

16.135 If there is a C to which A belongs but B does not, 
then A and B are not identical. 

Admittedly the last two laws are only hinted at. In another 
passage we find : 

16.14 For only to things that are indistinguishable and one 
in essence is it generally agreed that all the same attributes 
belong. 

This is almost the Leibnizian principium indiscernibilium in so 
many words, originating as we see with Aristotle. It is remarkable 
that we do not find the simple principle : 

16.141 If A is identical with B, and B with C, then A is 
identical with C. 



D. SYLLOGISMS FROM HYPOTHESES 

Aristotle did not know the expression 'hypothetical syllogism', 
but he often speaks of syllogisms from hypotheses (si* \)~o$£gz(x>c,) . 
We have shown above (in the commentary on citations 16.02 and 
16.03) that in general these need not be hypotheses; usually it is a 
matter only of logical laws or rules, similar to syllogisms in certain 
respects but not reducible to them. We have already seen some 
examples of such formulae. Here are some more which Aristotle 
would probably class with them. 

16.15 The refutation which depends upon the consequent 
arises because people suppose that the relation of consequence 
is convertible. For whenever, if this is, that necessarily is, they 
suppose that also when that is, this is. 

93 



THE GREEK VARIETY OF LOGIC 

16.16 When two things are so related to one another, that 
if the one is, the other necessarily is, then if the latter is not, 
the former will not be either, but if the latter is, it is not 
necessary that the former should be. 

16.17 If this follows that, it is claimed that the opposite 
of this will follow the opposite of that. . . . But that is not so; 
for the sequence is vice versa. 

16.18 In regard to subjects which must have one and one 
only of two predicates, as (e.g.) a man must have either 
disease or health, supposing we are well supplied as regards 
the one for arguing its presence or absence, we shall be well 
equipped as regards the remaining one as well. This rule is 
convertible for both purposes : for when we have shown that 
the one attribute belongs, we shall have shown that the 
remaining one does not belong; while if we show that the one 
does not belong, we shall have shown that the remaining one 
does belong. 

We have there, evidently, the exclusive alternative (negation of 
equivalence). 

16.19 In general whenever A and B are such that they 
cannot belong at the same time to the same thing, and one 
of the two necessarily belongs to everything, and again C and 
D are related in the same way, and A follows C but the 
relation cannot be reversed, then D must follow B and the 
relation cannot be reversed. And A and D may belong to the 
same thing, but B and C cannot. First it is clear from the 
following consideration that D follows B. For since either C 
or D necessarily belongs to everything; and since C cannot 
belong to that to which B belongs, because it carries A along 
with it and A and B cannot belong to the same thing; it is 
clear that D must follow B. Again since C does not reciprocate 
with A, but C or D belongs to everything, it is possible that A 
and D should belong to the same thing. But B and C cannot 
belong to the same thing, because A follows C; and so some- 
thing impossible results. It is clear then that B does not 
reciprocate with D either, since it is possible that D and A 
should belong at the same time to the same thing. 

This text is one of the peaks of Aristotelian logic: the founder of 
our science conducts himself with the same sureness and freedom 
as in the best parts of his syllogistic, though here dealing with a new 

94 



ARISTOTLE 

field, that of non-analytic formulae. The run of this text can be 
formulated thus: 

(1) For all X: A or B (and not both) belongs to X, and 

(2) for all X: C or D (and not both) belongs to X, and 

(3) for all X: if C belongs to X, then it belongs also to A. 
From these hypotheses there follows on the one hand : 

(4) for all X: if B belongs to X, then it belongs also to D, 
and on the other: 

(5) for all X: not both B and C belong to X. 

These consequences are quite correct. The thing to notice is that 
there are three different binary propositional functors ('or', 'and', 'if... 
then'). In his proof Aristotle uses, among others, the following three 
laws, apparently with full consciousness: 

16.191 For all X: if not both A and B belong to X, and B 
belongs to X, then A does not belong to it. 

16.192 For all X : if, when A belongs to X B also belongs to 
X, but B does not belong to X, then A also does not belong 
toX 

16.193 For all X: if either A or B belongs to X, and A does 
not belong to it, then B does belong to it. 



E. LAWS OF THE LOGIC OF RELATIONS 

16.20 If knowledge be a conceiving, then also the object 
of knowledge is an object of conception. 

16.21 If the object of conception is in some cases an object 
of knowledge, then also some form of conceiving is knowledge. 

16.22 If pleasure is good, then too a greater pleasure is a 
greater good ; and if injustice is bad, then too a greater injustice 
is a greater evil. 

In this connection the following piece of history deserves to go on 
record. De Morgan stated that the whole Aristotelian logic was 
unable to prove that if the horse is an animal, then the head of the 
horse is head of an animal. The reproach is evidently unjustified, 
since the law stated in 16.20 is just what is needed for this proof. 
Further, Whitehead and Russell (16.23) remark that the supposed 
lack of this law is really a good point abou£ Aristotelian logic, since 
it is invalid without an additional existential postulate. This may 
be right in relation to De Morgan's problem, i.e. if he understood 
'horse' as an individual name; but the law in which 16.20 is a 
substitution is correct - since it concerns not an individual but a 
class name ('knowledge'). 

95 



THE GREEK VARIETY OF LOGIC 

Aristotle gives three further laws of the logic of relations in the 
chapter about those syllogisms which later came to be called 
'obliqui' : 

16.24 That the first term belongs to the middle, and the 
middle to the extreme, must not be understood in the sense 
that they can always be predicated of one another. . . . But 
we must suppose the verb 'to belong' to have as many 
meanings as the senses in which the verb 'to be' is used, and 
in which the assertion that a thing 'is' may be said to be true. 
Take for example the statement that there is a single science 
of contraries. Let A stand for 'there being a single science', 
and B for things which are contrary to one another. Then A 
belongs to jB, not in the sense that contraries are the fact of 
there being a single science of them, but in the sense that it is 
true to say of the contraries that there is a single science of 
them. 

16.25 It happens sometimes that the first term is stated of 
the middle, but the middle is not stated of the third term, e.g. 
if wisdom is knowledge, and wisdom is of the good, the con- 
clusion is that there is knowledge of the good. The good then 
is not knowledge, though wisdom is knowledge. 

16.26 Sometimes the middle term is stated of the third, but 
the first is not stated of the middle, e.g. if there is a science of 
everything that has a quality, or is a contrary, and the good 
both is a contrary and has a quality, the conclusion is that 
there is a science of the good, but the good is not science, nor 
is that which has a quality or is a contrary, though the good is 
both of these. 

16.27 Sometimes neither the first term is stated of the 
middle, nor the middle of the third, while the first is sometimes 
stated of the third, and sometimes not: e.g. if there is a genus 
of that of which there is a science, and if there is a science of 
the good, we conclude that there is a genus of the good. But 
nothing is (there) predicated of anything. And if that of which 
there is a science is a genus, and if there is a science of the good, 
we conclude that the good is a genus. The first term then is 
predicated of the extreme, but in the premisses one thing is 
not stated of another. 

We have here four more relational laws ; of greater importance is the 
introductory remark that the so-called 'copula' need not be 'is' but 
can be replaced by some other relation. A further interesting fact is 

96 



ARISTOTLE 

that Aristotle presupposes inter alia the following law from the logic 
of classes (in 16.26): 

16.261 For all x: if x is A and /J, then x is A or B. 

The introductory remark admittedly only reveals an intuition 
that is undeveloped. Nor did Aristotle link it up with his own pene- 
trating thesis of the manifold structure of the sentence according to 
the diversity of the categories (11.15), so rising to a higher syste- 
matic unity. Nevertheless the text cited does contain the beginnings 
of a logic of relations. 

Finally we can collect from at least six places in the Topics a 
group of rules, totalling eighteen altogether, which perhaps are 
to be interpreted as belonging to the logic of relations. We give three 
of them, again concerned with 'more' : 

16.28 Moreover, argue from greater and less degrees. In 
regard to greater degrees there are four commonplace rules. 
One is : See whether a greater degree of the predicate follows a 
greater degree of the subject: e.g. if pleasure be good, see 
whether also a greater pleasure be a greater good. . . . Another 
rule is: If one predicate be attributed to two subjects; then 
supposing it does not belong to the subject to which it is the 
more likely to belong, neither does it belong where it is less 
likely to belong; while if it does belong where it is less likely 
to belong, then it belongs as well where it is more likely. . . . 
Moreover: If two predicates be attributed to two subjects, 
then if the one which is more usually thought to belong to the 
one subject does not belong, neither does the remaining 
predicate belong to the remaining subject; or. if the one which 
is less usually thought to belong to the one subject does be- 
long, so too does the remaining predicate to the remaining 
subject. 

F. PROPOSITIONAL RULES AND LAWS 

Finally we find in Aristotle four formulae belonging to the most 
abstract part of logic, namely, propositional logic. Two of them even 
contain propositional variables: 

16.29 If when A is, B must be, (then) when B is not. A 
cannot be. 

That these are propositional variables, Aristotle says expressly: 

16.30 A is posited as one thing, being two premisses taken 
together. 

97 



THE GREEK VARIETY OF LOGIC 

16.31 If, when A is, B must be, (then) also when A is 
possible, B must be possible. 

It is to be noted that these propositional variables permit sub- 
stitution only of quite determinate expressions, namely conjunctions 
of premisses suitable for an analytic syllogism. 

16.32 From true premisses it is not possible to draw a false 
conclusion, but a true conclusion may follow from false 
premisses, true, however, only in respect to the fact, not to the 
reason. 

That is not yet the scholastic principle ex falso sequitur quodlibel, 
but only the assertion that one can form syllogisms in which one or 
both premisses are false, the conclusion true. 

16.33 If the conclusion is false, the premisses of the argument 
(Xoyo?) must be false, either all or some of them. 

This rule underlies the indirect proof of syllogisms (cf. 14.201-202). 
Note that it is a rule, not a law, and formulated quite generally, 
without being limited to two premisses. 

SUMMARY 

Reviewing the logical doctrines of Aristotle as presented, we can 
state : 

1 . Aristotle created formal logic. For the first time in history we 
find in him: (a) a clear idea of universally valid logic law, though he 
never gave a definition of it, (b) the use of variables, (c) sentential 
forms which besides variables contain only logical constants. 

2. Aristotle constructed the first system of formal logic that we 
know. This consists exclusively of logical laws, and was developed 
axiomatically, even in more than one way. 

3. Aristotle's masterpiece in formal logic is his syllogistic. This is a 
system of term-logic consisting of laws, not rules. In spite of certain 
weaknesses it constitutes a faultlessly constructed system. 

4. Besides the syllogistic, Aristotle constructed other portions of 
term-logic, including an extremely complex modal logic, as well as a 
number of laws and rules which overstep the bounds of the syllo- 
gistic. 

5. At the end of his life Aristotle, in a few texts, succeeded in 
formulating even propositional formulae; but these, like the non- 
analytic formulae of term-logic, he did not develop systematically. 

6. Aristotelian logic, though formal, is not formalislic. It is lacking 
also in understanding of the difference between rules and laws, and 
the semantics remain rudimentary, in spite of the many works 
which Aristotle devoted to the subject. 

98 



THEOPHRASTUS 

It is no exaggeration to say that nothing comparable h;j.s been seen 
in the whole history of formal logic. Not only is Aristotle's logic, 
according to all our information, a completely new creation, but it 
has been brought even by him to a high degree of completeness. 
Since moreover Aristotle's most important writings - most impor- 
tant because they were the only complete logical works - survived 
the cultural catastrophe of Greece, it is no wonder that the huge body 
of doctrine they contain should have continued to fascinate nearly 
all logicians for more than two thousand years, and that the whole 
history of logic has developed along the lines traced out in advance 
by Aristotle's thought. 

That has not been harmless for the development of our science. 
Even in antiquity there was a school of logicians which introduced a 
new set of problems different from those posed by the logic of Ari- 
stotle. We have only fragments of their work, and the authority of 
the founder of logic was so great that the achievements of this 
school were not at all understood during the long period from the time 
of the Renaissance up to and including the nineteenth century. We 
must now concern ourselves with them, but first a brief word must 
be said about Aristotle's first disciple, Theophrastus. 



§17. THEOPHRASTUS 

Theophrastus of Eresos, Aristotle's chief disciple and leader of the 
Peripatetic school after the founder's death, has, in company with 
his less significant colleague Eudemus, an important place in the 
history of logic, and that in three respects. First, he developed various 
of his master's doctrines in such a way as to prepare the ground for 
the later 'classical' logic; secondly, he set his own quite different 
system in opposition to the Aristotelian modal iogic; thirdly, he 
developed a doctrine of hypothetical arguments which was a pre- 
paration for Megarian-Stoic logic. 

His very numerous works (17.01) have all perished save for some 
one hundred fragments. These, however, are enough to tell us that he 
commented on the most important of Aristotle's logical works ( 1 7.02 1 , 
and they give us some insight into his own logical thought. 



A. DEVELOPMENT AND ALTERATION OF VARIOUS DOCTRINES 

17.03 Speech having a twofold relation - as the philoso- 
pher Theophrastus has shown - one to the hearers, to whom it 
signifies something, the other to the things about which it 
informs the hearers, there arise in respect of the relation to the 
hearers poetics and rhetoric, ... in respect of that to the things, 

99 



THE GREEK VARIETY OF LOGIC 

it will be primarily the philosopher's business, as he refutes 
falsehood and demonstrates truth. 

We can see that this is a new semiotic, with stress on what is now 
called the 'pragmatic' dimension of signs. 

17.04 Theophrastus rightly calls the singular sentence 
determined, the particular undetermined. 

17.05 Alexander opines that 'not belonging to all' and 'to 
some not belonging' differ only in the expression, whereas 
Theophrastus's view is that they differ also in meaning: for 
'not belonging to all' shows that (something) belongs to 
several, 'to some not belonging' that (not belonging) to one. 

A more important thought is the following: 

17.06 Consequently Theophrastus says that in some cases, 
if the determination (of quantity) TupoaSiopLCT^o^) does not 
also stand with the predicate, opposites, contradictories, will 
be true, e.g. he says that 'Phanias possesses knowledge', 
'Phanias does not possess knowledge', can both be true. 

This is not a matter, as Theophrastus mistakenly supposed, of 
quantification of the predicate, which Aristotle had rejected (12.03), 
but of a quantification of both parts of a subject when there is a 
two-place functor (cf. 44.22 ff.). This structure was only later treated 
in detail (cf. 28.15 ff., 42.06, 42.22). We have here the first beginnings 
of it. 

17.07 In those premisses which potentially contain three 
terms, viz. those . . . which Theophrastus called xa-ra 7rpocrAY)tJnv 
(for these have three terms in a sense; since in (the premiss) 'to 
all of that to all of which B belongs, A also belongs' in the two 
terms A and B which are explicit there is somehow comprised 
the third of which B is predicated . . .): (these premisses) 
. . . seem to differ from categorical ones only in expression, as 
Theophrastus showed in his On Affirmation. 

17.08 But Theophrastus in On Affirmation treats 'to that 
to which B (belongs, there belongs also) A' as equivalent 
(feov Suva(X£V7]v) to 'to all of that to all of which B belongs, 
A (also belongs) (cf. 14.24). 

17.09 But Theophrastus and Eudemus have given a 
simpler proof that universal negative premisses can be con- 
verted. . . . They conduct the proof so: A belongs to no B. 
If it belongs to none, A must be disjoined (dbus^suxToci) and 

100 



THEOPHRASTUS 

separated (xs/wptdTat) from B. But what is disjoined is 
disjoined from something disjoined. Therefore B too is quite 
disjoined from A. And if this is so, it belongs to no A. 

This shows that Theophrastus takes a purely extensional view of 
the terms (cf. §36, E) - so much so that one is led by this text (as 
by 17.13) to think of a diagrammatic scheme such as Leibniz used 

(36.14). 

17.10 To these four (Aristotelian syllogisms of the first 
figure) Theophrastus added five others, which are neither 
perfect nor indemonstrable. 

We no longer have the relevant text. Alexander's explanations 
(17.11) show that these are the five: 

17.111 If A belongs to all B and B to all C, then too C 
belongs to some A (Baralipton). 

17.112 If A belongs to no B, and B to all C, then too C 
belongs to no A (Celanles). 

17.1 13 If A b longs to all B and B to some C, then too C be- 
longs to some A (Dabitis). 

17.114 If A belongs to all C and B to no C, then to some 
A C does not belong (Fapesmo). 

17.115 If A belongs to some B and B to no 67, then too to 
some A C does not belong (Frisemomorum). 

These are what were later called the 'indirect' syllogisms of the 
first figure, deduced by means of the Aristotelian rules (cf. § 13, D). 



B. MODAL LOGIC 

All the texts so far quoted contain developments of or - often 
questionable - improvements on the Aristotelian logic. The Theo- 
phrastan theory of modal syllogisms, on the other hand, is an 
entirely new system, set in fundamental oppositions, as it appears, 
to that of Aristotle. 

17.12 Hence Aristotle says that universal negative possible 
premisses are not convertible. But Theophrastus says that 
these too, like the other negatives, can be converted. 

17.13 But Theophrastus and Eudemus, as we have already- 
explained at the beginning, say that universal negative 
(possible premisses) can be converted, like universal negative 
assertoric and necessary ones. Their convertibility they prove 

101 



THE GREEK VARIETY OF LOGIC 

thus: if A possibly does not belong to all B, B also possibly 
does not (belong) to all A; for if A possibly does not belong 
to all B . . ., then A can be disjoined from all B; but if this is 
so, B also can be disjoined from A; and in that case B also 
possibly does not belong to all A. 

17.14 It is (according to Aristotle) a property of the 
possible to convert, i.e. the affirmations and negations con- 
cerning it follow on each other . . . but it should be known that 
this conversion of premisses is not valid in the school of 
Theophrastus, and they do not use it. For there is the same 
reason (1) for saying that the universal negative possible 
(premiss) is convertible, like the assertoric and necessary, and 
(2) (for saying) that affirmative possibles are not convertible 
into negatives. 

In brief: according to Theophrastus all laws of conversion for 
problematic sentences are exactly analogous to those for assertoric 
sentences; and the 'reason' of which Alexander speaks, can only be, 
so it would seem, that the modal doctrine of Theophrastus is based 
on one-sided possibility, while Aristotle's is based on two-sided. 

Similarly the second fundamental thesis of the Aristotelian system 
is also rejected: for Theophrastus the functor of modality must be 
thought of as determining the whole sentence, not just one or both 
of its arguments, i.e. it must be thought of as standing at the 
beginning of the sentence (cf. commentary on 15.13). 

17.15 But his companions who are with Theophrastus and 
Eudemus, deny this, and say that all formulae consisting of a 
necessary and an assertoric premiss, so constituted as to be 
suitable for syllogistic inference, yield an assertoric conclusion. 
They take that from the (principle according to which) in all 
(syllogistic) combinations the conclusion is similar to the last 
and weaker premiss. 

17.16 But Theophrastus, (in order to prove) that in this 
combination (auprXox?)) the conclusion yielded is not necessary, 
proceeds thus: Tor if B necessarily (belongs) to C and A does 
not necessarily belong to B, if one disjoins the not necessary, 
evidently, as B is disjoined (from A), C too will be disjoined 
from A : hence does not necessarily belong to it in virtue of 
the premisses. 

17.17 They prove that this is so by material means (= by 
substitutions) also. For they take a necessary universal 
affirmative or negative as major (premiss), an assertoric 

102 



THEOPHRASTUS 

universal affirmative as minor, and show that these yield an 
assertoric conclusion. Suppose that animal (belongs; to all 
man necessarily, but man belongs (simply) to all in motion : 
(then) animal will not necessarily belong to all in motion. 

Now the basis for Aristotle's permitting the drawing of a necessary 
conclusion from one necessary and one assertoric premiss, was 
precisely his idea of the structure of modal sentences. Theophrastue 
certainly does not reject this idea in his extant fragments, and per- 
haps was not fully aware of it. But in any case all that we have of his 
modal logic gives evidence of a system presupposing the rejection 
of the Aristotelian structure of modal sentences. 



C. HYPOTHETICAL SYLLOGISMS 

We have no text of Theophrastus that contains anything of his 
theory of hypothetical propositions. He seems to have treated of 
them, for he distinguished the meaning of zl and iizzi (17.18). Possibly 
too it was he who introduced the terminology for these propositions 
which Galen aseribes to the 'old Peripatetics' (17.19). However, we 
know that he developed hypothetical syllogisms: 

17.20 He (Aristotle) says that many syllogisms are formed 
on hypotheses. . . . Theophrastus mentions them in his 
Analytics, as do Eudemus and some others of his companions. 

According to Philoponus, both of them 'and also the Stoics' 
wrote 'many-lined' treatises about these syllogisms (17.21). In fact, 
however, the treatment of only one kind of these syllogisms is 
expressly attributed to Theophrastus, that namely which consists of 
'thoroughly (St' oXcov) hypothetical' syllogisms. 

17.22 However, the thoroughly hypotheticals are reduced to 
the three figures in another way, as Theophrastus has proved in 
the first book of the Prior Analytics. A thoroughly hypothe- 
tical syllogism is of this kind: If A, then B; if B, then C; if 
therefore A, then C. In these the conclusion too is hypotheti- 
cal; e.g. if man is, animal is; if animal is, then substance is: 
if therefore man is, substance is. Now since in these too there 
must be a middle term in which the premisses convene (for 
otherwise here also there cannot be a conclusive link), this 
middle will be positioned in three ways. For if one premiss 
ends with it and the other begins with it, there will be the 
first figure ; it will be in fact as though it was predicated of one 
extreme, subjected to the other. ... In this way of linking one 

103 



THE GREEK VARIETY OF LOGIC 

can take also the converse of the conclusion, in such a way that 
(C) is not the consequent (E7r6[xevov) but the antecedent 
(y]You(ji£vov), not indeed simply, but with opposition, since 
when a conclusion 'if A, then C" has been gained, there is 
gained a conclusion 'if not C, then not A\ 

If the premisses begin differently and end similarly, the 
figure will be the second, like the second (in the system) of 
categorical (syllogisms). . . . e.g. If man, then animal; if 
stone, then not animal; therefore if man, then not stone. . . . 

If the premisses begin similarly and end differently, the 
figure will be like the third . . . e.g. if A, then B ; if not A, then 
C; it will follow : therefore, if not jB, then C, or, if not 67, then B. 

The formulae contained in this text are presented in such a way 
that from them alone it is impossible to tell whether their variables 
are term- or propositional variables. However, the substitutions 
show that the former is the case. Hence we have no reason to ascribe 
any law of propositional logic to Theophrastus. Yet it is most 
probable that in developing Aristotle's hints about 'syllogisms from 
hypotheses' he prepared the way for the Megarian-Stoic doctrine. 

The formulae in the text just cited are worded as rules; but we 
do not know whether this wording is due to Theophrastus himself, 
or to Alexander and so mediately to the Stoics. 



104 



III. THE MEGARIAN-STOIC SCHOOL 



§ 18. HISTORICAL SURVKY 

A. THINKERS AND SCHOOLS 

18.01 Euclid originated from Megara on the Isthmus. . . . 
He occupied himself with the writings of Parmenides; his 
pupils and successors were called 'Megarians', also 'Eristics' 
and later 'Dialecticians'. 

18.02 The Milesian Eubulides also belongs among the 
successors of Euclid; he solved many dialectical subtleties, 
such as The Liar. 

18.03 Eubulides was also hostile to Aristotle and made 
many objections to him. Among the successors belonged 
Alexinus of Elis, a most contentious man, whence he gained 
the name 'Elenxinus' ('Refuter'). 

18.04 Among (the pupils) of Eubulides was Apollonius, 
surnamed Cronus, whose pupil Diodorus, the son of Ameinias 
of Iasus, was also called Cronus . . .. He too was a dialecti- 
cian. . . . During his stay with Ptolemy Soter he was challenged 
by Stilpo to solve some dialectical problems; but as he could 
not do this immediately . . ., he left the table, wrote a treatise 
on the problems propounded, and died of despondency. 

18.05 Stilpo, from the Greek Megara, studied under some 
pupils of Euclid; others say that he studied under Euclid 
himself, and also under Thrasymachus of Corinth, the friend 
of Icthyas. He surpassed the rest in inventiveness of argument 
and dialectical art to such an extent that well-nigh all Greece 
had their eyes on him and was fain to follow the Megarian school. 

18.06 He caught in his net Crates and very many more. 
Among them he captured Zeno the Phoenician. 

18.07 Zeno, the son of Mnaseas or Demeas, was born at 
Citium, a small Greek town on the island of Cyprus, where 
Phoenicians had settled. 

18.08 He was ... a pupil of Crates; some say that he also 
studied under Stilpo. 

18.09 He was assiduous in discussion with the dialectician 
Philo and studied with him; so that Philo came to be admired 
by the more youthful Zeno no less than his master Diodorus. 

105 



THE GREEK VARIETY OF LOGIC 

18.10 He also spent some time under Diodorus, . . . studying 
hard at dialectics. 

18.11 Kleanthes the son of Phanias was born at Assus . . . 
joined Zeno . . . and remained true to his teaching. 

18.12 Chrysippus the son of Apollonius from Soli or 
Tarsus . . ., was a pupil of Cleanthes. 

18.13 He became so famous as a dialectician, that it was 
generally said that if the gods were to use dialectic, it would be 
none other than that of Chrysippus. 

It was necessary to cite these extracts from the Lives and Opinions 
of Famous Philosophers of Diogenes Laertius, in order to counter a 
widespread error to the effect that there was a Stoic, but no Megarian 
logic. From the passages quoted it appears unmistakably that (a) 
the Megarian school antedated the Stoic, (b) the founders of the 
Stoa, Zeno and Chrysippus, learned their logic from the Megarians, 
Diodorus, Stilpo and Philo. And again (c) we know at least three 
Megarian thinkers of importance in the history of logic - Eubulides, 
Diodorus, and Philo - while only one can be named from the Stoa, 
viz. Chrysippus who can lay claim to practically no basically original 
doctrine, whereas each of the three Megarians conceived a definitely 
original idea. 

Admittedly the Megarian school seems to have died out by the 
close of the 3rd century b.c, whereas the Stoa continued to flourish. 
Also the adherents of the latter disseminated logic in many excellent 
handbooks with the result that people, as in Galen's time, spoke only 
of Stoic logic. The least that can justly be required of us is to speak 
of a Megarian-Stoic logic. Possibly the basic ideas should be attri- 
buted to the Megarians, their technical elaboration to the Stoics, 
but this is mere conjecture. 

The names and doctrinal influences recorded by Diogenes can be 
conveniently summarized in the following table : 



106 



MEGARIAN-STOIC LOGIC 

Euclid of Megara, pupil of Socrates, 

founder of the Megarian or 'dialectical' school 

(ca. 400 b.c.) 



Alexinus of Elis 
called 'Elenxinus' 



Eubulides 
of Miletus 
discoverer 
of The Liar 



I 



Icthyas 

| Thrasymachus 
friend of 
Icthyas 



Apollonius 
Cronus 

.1 

Diodorus 
Cronus of Iasus 
307 b.c. — 



Philo of Megara 



Stilpo of 

Megara 

(ca. 320 b.c.) 

i 

Zeno of Citium 
founder of 
the Stoa 
ca. 300 b.c. 

Cleanthes of Assus 

Chrysippus of Soli 
'second founder of 
the Stoa' 

281/78-208/05(7) b.c. 



B. PROBLEMS OF LITERARY HISTORY 

The conditions for investigation of the Megarian-Stoic logic 
are much less favourable than those for that of the logic of Aristotle 
or even Theophrastus. We have the essential works of Aristotle 
entire, and in the case of Theophrastus are in possession at least of 
fragments quoted by competent experts who are not absolutely 
hostile to the author they cite. But for Megarian-Stoic teaching we 
have to rely essentially on the refutations of Sextus Empiricus, an 
inveterate opponent. As B. Mates rightly says, it is as though we 
had to rely for a knowledge of R. Carnap's logic only on existen- 
tialist accounts of it. Fortunately Sextus, though no friend to the 
Stoics, was (in contrast to most existentialists) well acquainted with 
formal logic, which he opposed from his sceptical point of view. 
We can moreover control at least some of his reports by means of 
other texts. 

But still we have nothing but fragments. We can hardly doubt 
that the material to hand suffers from many gaps: for instance 
term-logic is almost completely missing, and it seems hardly likely 
that it was wholly unconsidered in the Stoa. 



107 



THE GREEK VARIETY OF LOGIC 

Another problem concerns the interpretation. Even in antiquity 
Stoic texts were often 'aristotelized', propositional variables taken 
for term-variables etc. A similar defect characterizes all modern 
historians of logic, Prantl most of all, who completely mistook the 
significance of this logic. Peirce was the first to see that it was a 
propositional logic, and J. Lukasiewicz did a lasting service in giving 
the correct interpretation. Now there is available a scientific 
monograph - a rarety in history of logic - by B. Mates. So in the 
present state of research it can be stated with some certainty that 
we are again in a position to understand this extremely interesting 
logic. 

C. ORIGIN AND NATURE 

In reading the Megarian-Stoic fragments one's first impression 
is that here is something different from Aristotelian logic: termino- 
logy, laws, the very range of problems, all are different. In addition 
we are confronted with a new technique of logic. The most striking 
differences are that the Megarian-Stoic logic is firstly not a logic of 
terms but of propositions, and secondly that it consists exclusively 
of rules, not of laws - as does the Prior Analytics. The question at 
once arises, what was the origin of this logic. 

The answer is complex. First of all one cannot doubt that the 
Megarians and Stoics, who as we have seen (cf. 18.03) found an only 
too frequent delight in refutation, had a tendency to do everything 
differently from Aristotle. Thus for example they introduce quite 
new expressions even where Aristotle has developed an excellent 
terminology. 

Yet it should not be said that their logical thought could have 
developed uninfluenced by Aristotle. On the contrary, they appear to 
have developed just those ideas which are last to appear in the 
Organon. We find, for instance, a more exact formulation of the 
rules which Aristotle used in axiomatizing the syllogistic, and him- 
self partially formulated. Nor can it be denied that they developed 
his theory of 'syllogisms from hypotheses', chiefly on the basis of 
the preparatory work of Theophrastus. And generally speaking 
they everywhere show traces of the same spirit as Aristotle's, only 
in a much sharper form, that spirit being the spirit of formalized 
logic. 

And that is not yet all. In many of his non-analytical formulae 
Aristotle depends directly on pre-Platonic and Platonic discussions, 
and this dependence is still greater in the case of the Megarian- 
Stoic thinkers. It often happens that they transmute these discus- 
sions from the language of term-logic into that of propositional 
logic, and one can understand how they, rather than Aristotle, 
came to do this on such a scale. Aristotle always remained at heart a 

108 



MEGARIAN-STOIC LOGIC 

pupil of Plato's, looking for essences, and accordingly asking 
himself the question: 'Does A belong to B?' But the Megarians start 

from the pre-Platonic question: 'How can the statement p be 
refuted?' Alexinus was called 'Hefuter', and all these thinkers 
continued to be fundamentally refuters in their logic. Which means 
that their basic problems were concerned with complete propositions, 
whereas Aristotle had his attention fixed on terms. The thorough 
empiricism too, to which the Stoics gave their allegiance, contributed 
to this difference. 

As to details, propositional logic originated with the Megarians 
and Stoics, the second great contribution made by the Greeks to 
logic, and just what was almost entirely missing from Aristotelian 
logic. Then, as already stated, they understood formal treatment in 
a formalistic way, and laid the foundations of an exact semantics 
and syntax. Misunderstood for centuries, this logic deserves recogni- 
tion as a very great achievement of thought. 

Unfortunately no means is available for us to pursue the historical 
development of Megarian-Stoic investigations; we can only consider 
what we find at the end of this development, which seems to have 
already come with Chrysippus. Within a hundred and fifty years 
Greek logic rose with unbelievable speed to the very heights of 
formalism. We now have to view these heights as already attained. 
Our presentation cannot be historical; it can only proceed systemati- 
cally. 



§ 19. CONCEPT OF LOGIC. SEMIOTICS. MODALITIES 

A. LOGIC 

19.01 They (the Stoics) say there is a threefold division of 
philosophical speech: one (part) is the physical, another the 
ethical, the third the logical. 

19.02 They compare philosophy to an animal, the logical 
part corresponding to the bones and sinews, the ethical to the 
fleshy parts, the physical to the soul. Or again to an egg, the 
logical (part) being the outside (=the shell). ... Or again to a 
fertile field. The fence then corresponds to the logical. 

19.03 According to some the logical part is divided into 
two sciences, rhetoric and dialectic. . . . They explain rhetoric 
as the science of speaking well . . . and dialectic as the science 
of right discussion in speech, by question and answer. Hence the 
following definition: it is the science of the true, the false, and 
of what is neither of the two. 

109 



THE GREEK VARIETY OF LOGIC 

That of course does not mean that the Stoics knew of a three- 
valued logic (cf . 49.08) ; they refer only to sentences (which are true 
or false) and their parts (which are neither). - The text cited expres- 
ses the attitude of the Stoics to the problem of the place of logic 
among the sciences : for them it is quite unmistakably a part of the 
system. What more is said seems to concern a methodology of discus- 
sion rather in the manner of the Aristotelian Topics (11.01). But as 
we know from other fragments, it is only a consequence of the Stoic 
doctrine of the principal subject-matter of logic which consists in 
lecia (Xextoc). This important notion requires immediate clarifica- 
tion. 



B. LECTA 

19.04 The Stoics say that these three are connected: the 
significate (<yyj[jiaiv6[ji£vov), the sign (<n)(Jiaivov) and the thing 
(tuyxocvov). The sign is the sound itself, e.g. the (sound) 'Dion', 
the significate is the entity manifested by (this sign) and which 
we apprehend as co-existing with our thought, (but) which 
foreigners do not comprehend, although they hear the sound; 
the thing is the external existent, e.g. Dion himself. Of these, 
two are bodies, viz. the sound and the thing, and one imma- 
terial, viz. the entity signified, the lecton, which (further) is 
true or false. 

19.05 They say that the lecton is what subsists according 
to a rational presentation (xara cpavTacuav XoyixYjv). 

19.06 Some, and above all those of the Stoa, think that 
truth is distinguished in three ways from what is true, . . . 
truth is a body, but what is true is immaterial; and this is 
shown, they say, by the fact that what is true is a proposition 
(a££<o[i.a), while a proposition is a lecton, and lecia are imma- 
terial. 

We have refrained from translating the Greek expression Xsxtov 
which derives from Xsyeiv and literally means 'what is said', i.e. 
what one means when one speaks meaningfully. The text last cited, 
about truth and what is true, is to be specially noted. The former is 
something psychic, and for the Stoics all such, every thought in 
particular, is material. But the lecton is not a quality of the mind, or 
in scholastic terminology a conceptus subjectivus. To use Frege's 
language it is the sense (Sinn) of an expression, scholastically the 
conceptus objectivus, what is objectively meant. In the (pseudo-) 
Aristotelian Categories there is a passage (10.29) about the Xoyo? tou 
7rpay(jLaTO^, which corresponds to the Stoic lecton. Only in the Stoa 

110 



MEGARIAN-STOIC LOGIC 

the leclon has become the chief subject-matter of logic and indeed 

the unique subject of formal logic. That certainly jettisons the 
Aristotelian neutrality of logic arid supposes a definite philosophical 
standpoint. But the original philosophical intuition involved is to 
be the more noticed in that very many philosophers and logicians, up 
to the most recent times, have confused the leclon with psychic 
images and occurrences (cf. 26.07, 36.08). That the Stoic logic is a 
science of lecta is made plain by their division : 

19.07 Dialectic is divided, they say, into the topic of 
signiflcates and (the topic) of the sound. That of significates 
is divided into the topic of conceptions, and that of the lecta 
which co-exist with them: propositions, independents (lecta), 
predicates, and so on . . ., arguments and moods and syllogisms, 
and fallacies other than those arising from the sound and the 
things. ... A topic proper to dialectic is also that already 
mentioned about the sound itself. 



C. SYNTAX 

19.08 The elements of speech are the twenty-four letters. 
But 'letter' can have three meanings: the letter (itself), the 
(written) sign (^apaxr/jp) of the letter, and its name, e.g. 
'alpha'. . . . Utterance (<pcoWj) is distinguished from locution 
(Xe£is) in that a mere sound is utterance, but only articulated 
sound is locution. Locution is distinguished from speech in 
that speech is always meaningful, while what has no meaning 
can be locution, e.g. 'blityri' - which is not speech. 

19.09 There are five parts of speech, as Diogenes, in his 
(treatise) On Utterance, and Ghrysippus say: proper names 
(ovofi-a), general names (7rpo<77)Yopia), verbs (pjfxa), connections 
(ouvSsctu,o?), articles. . . . 

19.10 A general name is according to Diogenes a part of 
speech which signifies a common quality, e.g. 'man', 'horse'. 
But a proper name is a part of speech which manifests a 
quality proper to one, e.g. 'Diogenes', 'Socrates'. A verb is 
according to Diogenes a part of speech, which signifies an 
incomposite predicate (xaTYjyopYjjjia), or as others (define it), an 
indeclinable part of speech which signifies something co- 
ordinated with one or more, eg. 'I write', 'I speak'. A con- 
junction is an indeclinable part of speech which connects its 
parts. 

Ill 



THE GREEK VARIETY OF LOGIC 

Accounts of the division of lecla contradict one another, and are 
obscure. The following scheme composed by B. Mates (19.11) may 
best correspond to the original Stoic teaching: 

What is said 

Xextov 



incomplete complete 

zKknzic, auTOTsXli; 



predicate subject proposition others 

xaT7]YOp7](jLa titcoctk; a£ia)(j,a 7ttK7[ia 

But the division of propositions is clearly and fully transmitted. 

19.12 A proposition is what is true or false, or a complete 
entity (Trpayfxa) assertoric by itself, e.g. 'It is day*, 'Dion walks 
about'. It is called 'axiom' (a£ico[xa) from being approved 
(a£iouc7&ai) or disapproved. For he who says 'it rs day' seems to 
admit that it is day; and when it is day, the foregoing axiom is 
true ; but when it is not (day), false. Different from one another 
are axiom, question, inquiry, command, oath, wish, exhorta- 
tion, address, entity similar to an axiom. 

19.13 Of axioms, some are simple, some not simple, as is 
said in the schools of Chrysippus, Archedemus, Athenodorus, 
Antipater and Crinis. Simple are those which consist of an 
axiom not repeated (fjrf) Swccpopou^evou), e.g. 'it is day'. Ones not 
simple are those consisting of a repeated axiom or of more 
than one axiom. An example of the former is: 'if it is day, it is 
day' ; of the latter : 'if it is day, it is light' . 

19.14 Of simple axioms some are definite (cbpLcjjjiva), others 
indefinite, others again intermediate (fxicja). Definite are those 
which are referentially expressed, e.g. 'this man walks about', 
'this man sits' : (for they refer to an individual man). Indefinite 
are those in which an indefinite particle holds the chief place 
(xupLEuei), e.g. 'someone sits'. Intermediate are those such as: 
' a man sits' or 'Socrates walks about'. . . . 

19.15 Among axioms not simple is the compound (cjuvyj^fiivov 
= conditional), as Chrysippus in the Dialectic and Diogenes in 
the Dialectic AH say, which is compounded by means of the 
implicative connective 'if ; this connective tells one that the 
second follows from the first, e.g. 'if it is day, it is light'. An 
inferential axiom (7rapaauvy)[i.(xevov) is, as Crinis says in the 
Dialectic Art, one which begins and ends with an axiom and is 

112 



MEGARIAN-STOIC LOGIC 

compounded 7capamwJ7CTai) by means of the connective 'since 1 
(inei), e.g. 'since it is day, it is light'. This connective tells one 
that the second follows from the first and that the first is the 
case. Conjunctive (au[jwus7cXeYjjivov) is the axiom compounded by 
means of a conjunctive connective, e.g. 'it is day and it is 
light'. Disjunctive (Sis^euytiivov) is the axiom compounded by 
means of the separative connective 'or', e.g. 'it is day or it is 
night'. This connective tells one that one of the axioms is 
false. Causal (amto<k<;) is the axiom compounded by means 
of the connective 'because', e.g. 'because it is day, it is light'. 
For it is here to be understood that the first is the cause of the 
second. An axiom showing what is rather the case is one com- 
pounded by means of the connective 'rather than' which 
shows this and stands in the middle of the axiom, e.g. 'it is 
night rather than day'. 

Note in these texts that lecla, not words or psychic events are the 
subject-matter throughout. Hence most translations (those e.g. of 
Apelt, 19.16, and Hicks) are misleading, since they talk of connective 
'words' and 'judgements'. 



D. DOCTRINE OF CATEGORIES 

19.17 The common genus 'what is' has nothing over it. It 
is the beginning of things and everything is inferior to it. The 
Stoics wanted to put another, still more principal genus above 
it. 

19.18 To some Stoics 'what' seems to be the prime genus; 
and I will say why. In nature, they say, some things exist, 
others do not. Even those which do not exist are contained in 
nature, those which occur in the soul, like centaurs, giants 
and anything else which acquires an image when falsely 
framed in thought, though having no substance. 

So according to these Stoics there is a summum genus. This is a 
regression in comparison with Aristotle's subtle anticipation of a 
theory of types (11.16). 

19.19 But the Stoics think that the prime genera are more 
limited in number (than the Aristotelian). . . . For they 
introduce a fourfold division into subjects (u7rox£iji.sva), qualia 
(tcoloc), things that are in a determinate way (ttco; £'x 0VTa )> and 
things that are somehow related to something (r.pbc, ti -co; 
£XOVTa). 

113 



THE GREEK VARIETY OF LOGIC 

These four categories are not to be understood as supreme genera 
(under the 'what'): That is, it is not the case that one being is a 
subject, another a relation, but all the categories belong to every 
being, and every category presupposes the preceding ones (19.20). 
This doctrine has no great significance for logic. 



E. TRUTH 

Apart from the distinction already mentioned between what is 
true and truth (19.06), the Stoics seem to have used the word 'true' 
in at least five senses. On this Sextus says : 

19.21 Some of them have located the true and the false in 
the significates ( =leda), others in the sound, others again in 
the operation of the mind. 

As regards the truth of lecla, a further threefold distinction can be 
made between: 

1. truth of propositions. 

2. truth of propositional forms (i.e. what it is that sentential 
functions refer to). That the Megarians and Stoics attributed truth 
and falsity to such propositional forms is seen in their teaching 
about functors (vide infra). 

3. truth of arguments (vid. 21.07). 

Those are all lecla, but Sextus refers to two further kinds of truth: 

4. truth of ideas (19.22). 

5. truth of sentences. 

According to all our information the first kind of truth was 
fundamental, as presupposed in all the others. Thus for instance the 
Stoics defined the truth of propositional forms by its means, with 
the help of time-variables; the truth of arguments in terms of the 
truth of the corresponding conditional propositions; while the truth 
of ideas and sentences is similarly reducible to that of lecla, accord- 
ing to what we know of the relation between them. 



F. MODALITIES 

Only fragments have come down to us of the very interesting 
Megarian doctrine of modalities. It seems to be an attempt to reduce 
necessity and possibility to simple existence by means of time- 
variables, a proceeding wholly consonant with the empirical stand- 
point of these thinkers. We give only the two most important 
passages on the subject: 

19.23 'Possible' can also be predicated of what is possible in 
a 'Diodorean' sense, that is to say of what is or will be. For he 

114 



MEGAR IAN-STOIC LOGIC 

(Diodorus) deemed possible only what either is or will be. 
Since according to him it is possible that I am in Corinth, if 
I am or ever shall be there; and if I were not going to be 
there, it would not be possible. And it is possible that a child 
should be a grammarian, if he will ever be one. To prove this, 
Diodorus devised the master-argument (xupiefoov). Philo took a 
similar view. 

19.24 The (problem of) the master-argument seem- prob- 
ably to have originated from the following considerations. As 
the following three (propositions) are incompatible: (1) 
Whatever is true about the past is necessary, (2) the impos- 
sible does not follow from the possible which neither is nor 
will be true - Diodorus, comparing this incompatibility with 
the greater plausibility of the first two, inferred that nothing is 
possible which neither is nor will be true. 

Unfortunately that is the only really explicit text about the 
celebrated master-argument of Diodorus. It fails to enable us to 
survey the whole problem, because we do not know why the three 
propositions should be incompatible. One thing seems clear: that 
possibility was defined in the following way : 

19.241 p is (now) possible if and only if p is now true or 
will be true at some future time. 

From a rather vague text of Boethius (19.25) we further learn 
that the definitions of the other possibility-functors must be more 
or less as follows : 

19.242 p is (now) impossible if and only if p is not true and never 
will be true. 

19.243 p is (now) necessary if and only if p is true and always will 
be true. 

19.244 p is (now) not necessary if and only if p is not true or will 
not be true at some future time. 



§20. PROPOSITIONAL FUNCTORS 

To the credit of the Megarian-Stoic school are some very subtle 
researches into the most important propositional functors. The 
thinkers of the school even succeeded in stating quite correct 
truth-matrices. 

115 



THE GREEK VARIETY OF LOGIC 
A. NEGATION 

20.01 Negative are said to be only those propositions to 
which the negative particle is prefixed. 

This text shows something to which many passages bear witness, 
that the Stoics constructed their logic not merely formally, but quite 
formalistically. This was blamed by Apuleius (20.02) and Galen 
(20.03), who said that the Stoics were only interested in linguistic 
form. But this reproach - if indeed it is one - cannot be sustained 
in view of what we know of the subject-matter of Stoic logic; Stoic 
formalism is concerned with words only as signs of lecta. 

20.04 Among simple axioms are the negative (darocpaTLxov), 
the denying (apv7)Ti.x6v), the privative ((jTepyraxov). . . . An 
example of the negative is: 'it is not day'. A species of this 
is the super-negative (u7repoc7rocpaT[.x6v). The super-negative is 
the negation of the negative, e.g. 'not - it is not day'. This 
posits 'it is day'. A denying (axiom) is one which consists of 
a negative particle and a predicate, e.g. 'No-one walks about'. 
Privative is one which consists of a privative particle and 
what has the force of an axiom, e.g. 'this man is unfriendly 
to man'. 

The extant fragments do not contain a table of truth-values for 
negation, but the text cited evidently contains the law of double 
negation : 

20.041 not-not p if and only if p (cf. 24.26). 



B. IMPLICATION 

The definition of implication was a matter much debated among 
the Megarians and Stoics: 

20.05 All dialecticians say that a connected (proposition) is 
sound (uyti<;) when its consequent follows from (axoAoufrsi) its 
antecedent - but they dispute about when and how it follows, 
and propound rival criteria. 

Even so Callimachus, librarian at Alexandria in the 2nd century 
b.c, said: 

20.06 The very crows on the roofs croak about what 
implications are sound. 

116 



MEGARIAN-STOIC LOGIC 

1. Philonian implication 

20.07 Philo said that the connected (proposition) is true 
when it is not the case that it begins with the true and ends 
with the false. So according to him there are three ways in 
which a true connected (proposition) is obtained, only one in 
which a false. For (1) if it begins with true and ends with true, 
it is true, e.g. 'if it is day, it is light' ; (2) when it begins with false 
and ends with false, it is true, e.g. 'if the earth flies, the earth 
has wings'; (3) similarly too that which begins with false and 
ends with true, e.g. 'if the earth flies, the earth exists'. It is 
false only when beginning with true, it ends with false, e.g. 
'if it is day, it is night' ; since when it is day, the (proposition) 
'it is day' is true - which was the antecedent ; and the (proposi- 
tion) 'it is night' is false, which was the consequent. 

Here some terminological explanations are required. The Stoics 
called the antecedent "/jyouuxvov, the consequent Xvjyov, and moreover 
had the corresponding verbs: ^yetTOu, Xyjyei, untranslatable in their 
technical use. Hence we have simply translated these words accord- 
ing to their ordinary sense, by 'begins' and 'ends'. The term too for 
the sentences themselves (or the propositions to which they refer) has 
been translated according to its everyday sense by 'connected', the 
word 'conditional' having been avoided because apparently the 
idea of condition was foreign to Megarian-Stoic thought. 

As to the content of the passage, it gives us a perfect truth- 
matrix, which can be set out in tabular form thus: 

20.071 



Antecedent 


Consequent 


Connected 
proposition 


true 


true 


true 


false 


false 


true 


false 


true 


true 


true 


false 


false 



It is, as we can see, the truth-value matrix for material impli- 
cation, ordered otherwise than is usual nowadays (41.12; but 42.27). 
The latter therefore deserves to be called 'Philonian'. 

2. Diodorean implication 

20.08 Diodorus says that the connected (proposition) is 
true when it begins with true and neither could nor can end 
with false. This runs counter to the Philonian position. For the 
connected (proposition) 'if it is day, I converse' is true 
according to Philo, in case it is day and I converse, since it 

117 



THE GREEK VARIETY OF LOGIC 

begins with the true (proposition) 'it is day' and ends with the 
true (proposition) 'I converse'. But according to Diodorus (it is) 
false. For at a given time it can begin with the true (proposi- 
tion) 'it is day' and end with the false (proposition) ' I converse' , 
suppose I should fall silent . . . (and) before I began to con- 
verse it began with a true (proposition) and ended with the 
false one 'I converse'. Further, the (proposition) 'if it is 
night, I converse' is true according to Philo in case it is day 
and I am silent; for it (then) begins with false and ends with 
false. But according to Diodorus (it is) false; for it can begin 
with true and end with false, in case the night is past and 
I am not conversing. And also the (proposition) 'if it is night, 
it is day' is according to Philo true in case it is day, because, 
while it begins with the false (proposition) 'it is night', it ends 
with the true (proposition) 'it is day'. But according to Dio- 
dorus it is false because, while it can begin - when night is 
come - with the true (proposition) 'it is night', it can end with 
the false (proposition) 'it is day'. 

So we can fix Diodorean implication by the following definition : 

20.081 If p, then q, if and only if, for every time I it is 
not the case that p is true at t and q is false at t. 

3. 'Connexive'' implication 

20.09 (According to Diodorus) this (proposition) is true: 
'if there are no atomic elements of things, then there are 
atomic elements of things' . . . but those who introduce con- 
nection ((TuvapTTjcjLv) say that the connected (proposition) is 
sound when the contradictory (avTt,x£i[i.s:vov) of its consequent 
is incompatible ((xdcx^Tai) with its antecedent. So according to 
them the aforesaid connected (propositions) (20.07) are bad 
({AoxOvjpa), but the following is true (<xXyfi£q) : 'if it is day, it is 
day'. 

20.10 A connected (proposition) is true in which the opposite 
of the consequent is incompatible with the antecedent, e.g. 
'if it is day, it is light'. This is true, since 'it is not light', the 
opposite of the consequent, is incompatible with 'it is day'. 
A connected (proposition) is false in which the opposite of the 
consequent is not incompatible with the antecedent, e.g. 'if it 
is day, Dion walks about'; for 'Dion is not walking about' is 
not incompatible with 'it is day'. 

118 



MEGARIAN-STOIC LOGIC 

This definition is often ascribed to Chrysippus (20.11), but that 
it originated with him may be doubted (20.12). It is not clear how- 
it is to be understood. Perhaps we have here an ancient form of 
strict implication (49.04 31.13). 

4. ' Inclusive' implication 

20.13 Those who judge (implication) by what is implicit 
(eu-cpaaei xptvovT£<;) say that the connected (proposition) is 
true when its consequent is potentially (Suvdqiei) contained in 
the antecedent. According to them the (proposition) 'if it is 
day, it is day' and every repetitive connected (proposition) is 
probably false, since nothing can be contained in itself. 

This definition too is not now fully intelligible. It seems to concern 
a relation of subordination something like that which holds between 
a statement about all elements of a class and one about the elements 
of one of its sub-classes. No further reference to this definition is to 
be found in our sources; perhaps it was only adopted by isolated 
logicians of the school. 

C. DISJUNCTION 

We know much less about disjunction than about implication. 
Apparently it formed the subject of the same sort of dispute that 
there was about the definition of implication. But our texts are few 
and obscure. It is only certain that two kinds of disjunction were 
recognized : the complete (exclusive) and the incomplete (not exclu- 
sive), of which the first is well exemplified. 

1. Complete disjunction 

20.14 The disjunctive (proposition) consists of (contra- 
dictorily) opposed (propositions), e.g. of those to the effect 
that there are proofs and that there are not proofs. . . . For as 
every disjunctive is true if (and only if) it contains a true 
(proposition) and since one of (two contradictorily) opposed 
(propositions) is evidently always true, it must certainly be 
said that the (proposition) so formed is true. 

20.15 There is also another (proposition) which the Greeks 
call Sis^euyuivov a£icou.a and we call disjunctum. This is of the 
kind : 'pleasure is either good or bad, or neither good nor bad'. 
Now all (propositions) which are disjoined (disjuncta) (within 
one such proposition) are mutually incompatible, and their 
opposites, which the Greeks call avn.xsiu.sva must also be 
mutually opposed (contraria). Of all (propositions) which are 
disjoined, one will be true, the others false. But when none of 

119 



THE GREEK VARIETY OF LOGIC 

them at all are true, or all, or more than one are true, or when 
the disjoined (propositions) are not incompatible, or when 
their opposites are not mutually opposed, then the disjunctive 
(proposition) will be false. They call it TOxpocSis^euyfiivov. 

20.16 The true disjunctive (proposition) tells us that one of 
its propositions is true, the other or others false and incompat- 
ible. 

These texts offer a difficulty, in the supposition that a statement 
can be contradictory to more than one other. However, the practice 
of the school concerning the disjunction here defined is clear: in the 
sense envisaged 'p or q' is understood as the negation of equivalence 
(vide infra 22.07), i.e. in such a way that just one of the two argu- 
ments is true and just one false. 

2. Incomplete disjunction 

The surviving information about this is very vague. The best is 
given by Galen, but raises the question how much of it is Megarian- 
Stoic doctrine and how much Galen's own speculation: 

20.17 This state of things exhibits a complete incompat- 
ibility (tiXeiav paxyjv), the other an incomplete (sXXottjv) 
according to which we say for example: 'if Dion is at Athens, 
Dion is not at the Isthmus'. For this is characteristic of in- 
compatibility, that incompatibles cannot both be the case; 
but they differ in that according to the one the incompatibles 
can neither both be true nor both false, but according to the 
other this last may occur. If then only inability to be true 
together characterizes them, the incompatibility is incomplete, 
but if also inability to be false together, it is complete. 

20.18 There is no reason why we should not call the propos- 
ition involving complete incompatibility 'disjunctive' and that 
involving incomplete incompatibility 'quasi-disjunctive'. . . . 
But in some propositions not only one, but more or all 
components can be true, and one must be. Some call such 
'sub-disjunctive' (7cap<x8is£euY(Aeva) ; these contain only one 
true (proposition) among those disjoined, independently of 
whether they are composed of two or more simple propositions. 

Evidently this is a matter of two different kinds of disjunctive 
propositions, and so of disjunction. The first is called 'quasi-disjunc- 
tion' and seems to be equivalent to the denial of conjunction: 

20.181 p or q if and only if, not: p and q. 
120 



MEGARIAN-STOIC LOGIC 

Then the intended functor would be that of Shelter (43.43). 
The second kind is called 'sub-disjunctive' and could be defined 
by the following equivalence: 

20.182 p or (also) q if and only if: if not p, then q. 

This is the modern functor of the logical sum (cf. 14.1011. . 

Neither of these two functors was used by the Stoics in practice, 
at least so far as we can ascertain from the extant sources. 

D. CONJUNCTION 

20.19 What the Greeks call (jufJwue7cXeYjjtivov we call conjuncium 

or copulatum. It is as follows : 'Publius Scipio, son of Paulus, was twice 
consul and had a triumph and was censor and was colleague of 
Lucius Mummius in the censorship.' In every conjunctive the whole 
is said to be false if one (component) is false, even if the others are 
true. For if I were to add to all that I have truly said about that 
Scipio: 'and overcame Hannibal in Africa', which is false, then the 
whole conjunctive which includes that would be false: because 
that is a false addition, and the whole is stated together. 

E. EQUIVALENCE 

20.20 Syllogisms which have hypothetical premisses are 
formed by transition from one thing to another, because of 
consequence (axoXou0La) or incompatibility, each of which 
may be either complete or incomplete. 

20.21 The (exclusive) disjunctive premiss (Siaiperwdj) is 
equivalent to the following: 'if it is not day, it is night'. 

This last cited text, in which quite certainly complete disjunction 
is intended (20.14 ff.) can only be understood as referring to 'com- 
plete consequence' (20.20) - and then we have equivalence. In this 
case we have the following definition, in which 'or' is to be under- 
stood in the exclusive sense: 

20.211 q completely follows from p if and only if, not : p or q. 

We owe the discovery of these facts to Stakelum (20.22). Boethius, 
probably drawing on a Stoic source, understands 'if A - B' in just 
this sense 20.23). So it can be taken as likely that the functor of 
equivalence was known to the Stoics as 'complete consequence'. 

F. OTHER FUNCTORS 

We also have definitions of the inferential proposition (cf. 19.15 . 
This consists of a combination of conjunction with Diodorean 
(certainly not Philonian) implication. Other kinds of compound 

121 



THE GREEK VARIETY OF LOGIC 

propositions are the causal and the relative; their functors are not 
definable by truth-matrices. Possibly there are further functors of 
similar nature. 



§21. ARGUMENTS 
AND SCHEMES OF INFERENCE 

A. CONCLUSIVE, TRUE, AND DEMONSTRATIVE ARGUMENTS 

21.01 An argument (Xoyo^) is a system of premisses and 
conclusion. Premisses are propositions agreed upon for the 
proof of the conclusion, the conclusion is the proposition 
proved from the premisses. E.g. in the following (argument): 
'if it is day, it is light; it is day; therefore it is light', 'it is 
light' is the conclusion, the other propositions are premisses. 

21.02 Some arguments are conclusive (cjuvocxtixol), others 
not conclusive. They are conclusive when a connected 
proposition, beginning with the conjunction of the premisses 
of the argument and ending with the conclusion, is true. 
E.g. the argument mentioned above is conclusive, since from 
the conjunction of its premisses 'if it is day, it is light' and 'it is 
day' there follows 'it is light' in this connected proposition: 
'if: it is day and if it is day, it is light: then it is light.'* Not 
conclusive are arguments not so constructed. 

This is a very important text, showing how accurately the Stoic 
distinguished between a conditional proposition and implication 
on the one hand, and an argument or inferential scheme and the 
consequence-relation on the other. For an argument is conclusive 
(cruvaxTLxo^) when the corresponding conditional proposition is true 

The Stoics had a set terminology for the components of an argu- 
ment. In the simplest case it has two premisses, X^jJiaTa (in the 
wider sense); the first is also called XyjfjLfxa (the narrower sense), in 
contrast to the second which is called izpoa'krityic, 21.04); when the 
first premiss is connected, it is also called Tpomxov (21.05). 

21.06 Of arguments, some are not conclusive (aTuepavToi), 
others conclusive (nzpoLvrixoi) . Not conclusive are those in 
which the contradictory opposite of the conclusion is not 

Reading etnep el rjjjipa sax£, xal ■yjjxepa eari, cpcot; ecrdv. This reading was 
called a 'monstrosity' by Heintz, whereas it is evidently the only correct one 
(21.03). 

122 



MEGA R IAN-STOIC LOGIC 

incompatible with the conjunction of the premisses, e.g. such 
as: 'if it is day, it is light; it is day; therefore Dion walks 
about'. 

It seems to follow that the conditional sentence corresponding 
to an argument must contain the functor of connexive implication 
for an argument to be conclusive (cf. 20.09 f. and the commentary). 

21.07 Of conclusive arguments some are true, others false. 
They are true when besides the connected proposition, which 
consists of the conjunction of the premisses and the conclusion, 
being true, the conjunction of the premisses is also* true, i.e. 
that which forms the antecedent in the connected proposition. 

Again a text of the utmost importance, expressing a clear distinc- 
tion between formal validity and truth. This distinction was admit- 
tedly known to Aristotle (10.05 f.), but this is the first explicit 
accurate formulation. 

21.09 Of true arguments some are demonstrative (<x7co&eix- 
tlxol) others not demonstrative. Demonstrative are those con- 
cluding to the not evident from the evident, not demonstrative 
are those not of that kind. E.g. the argument: 'if it is day, it is 
light; it is day; therefore it is light' is not demonstrative, for 
that it is light (which is evident) is its conclusion. On the other 
hand, this is demonstrative: 'if the sweat flows through the 
surface, there are intelligible (voyjtol) pores; the sweat flows 
through the surface; therefore there are intelligible pores', for 
it has a non-evident conclusion, viz. 'therefore there are 
intelligible pores'. 

'Intelligible' here means 'only to be known by the mind'; the 
pores are not visible. 

The division of arguments comprised in this last series of texts is 
logically irrelevant, but of great methodological interest. It can be 
presented thus: 



[ 


demonstrative 


[ true I 




( conclusive I { 


not demonstrative 


arguments I [ not true 




( not conclusive 




* Omitting with Mates (21.08): xal to G\j[iniptxo\±ot.. 





123 



THE GREEK VARIETY OF LOGIC 
B. NON-SYLLOGISTIC ARGUMENTS 

A further interesting division shows how accurately the Stoics 
distinguished between language and meta-language : 

21.10 Of conclusive arguments, some are called by the name 
of the genus, 'conclusive' (rapavTLxoL), others are called 'syl- 
logistic'. Syllogistic are those which are either indemonstrable 
(ava7r6<teixTOL) or are reduced to the indemonstrable by means 
of one or more rules (tc5v 0£u.dcTcov), e.g. 'if Dion walks about, 
Dion is in motion; Dion walks about; therefore Dion is in 
motion'. Conclusive in the specific sense are those which do not 
conclude syllogistically, those of e.g. the following kind : 'it is 
false that it is day and it is night; it is day; therefore it is not 
night'. Non-syllogistic, on the other hand, are arguments 
which appear to resemble syllogistic ones, but do not conclude, 
e.g. 'if Dion is a horse, Dion is an animal; Dion is not a horse; 
therefore Dion is not an animal'. 

21.11 . . . but the moderns, who follow the linguistic 
expression, not what it stands for, . . . say that if the expres- 
sion is formulated thus: 'if A, then B; A; therefore B,' the 
argument is syllogistic, but 'B follows on A; A; therefore B' 
is not syllogistic, though it is conclusive. 

21.12 . . . The kind of argument which is called 'unmetho- 
dically concluding' (au^OoSox; 7cepaivovTes) is e.g. this: 'it is 
day; but you say that it is day; therefore you say true'. 

21.13 (Those which the moderns call 'unmethodically 
concluding' . . .) are such as the following: 'Dion says that it is 
day; Dion says true; therefore it is day'. 

21.14 . . . like the unmethodically concluding arguments 
among the Stoics. When e.g. someone says: 'the first (is) 
greater than the second, the second than the third, therefore 
the first (is) greater than the third'. 



C. FURTHER KINDS OF ARGUMENT 

21.15 Those arguments too which they call 'duplicated' 
(SioccpopoVevoi) are n °t syllogistic, e.g. this: 'if it is day, it is 
day; therefore it is day'. 

21.16 The argument): 'if it is day, it is light; it is day; 
therefore it is day', and in general those which the moderns 
call 'not diversely concluding' (aSt^opo^ rcepaivovTe^) . . . . 

124 



M E G A R I A N - S T O I C LOGIC 

21.17 Antipater, one of the most celebrated men of the 
Stoic sect, used to say that arguments with a single premiss 
can also be formed (|jiovoXy)(IU,<xtol). 

21.18 From one premiss there results no (conclusive) com- 
bination (colledio), though the consequence (conclusio) 'you 
see, therefore you live' seemed complete to Antipater the 
Stoic, against the doctrine of all (others) - for it is complete 
(only) in the following way; 'if you see, then you live; you see; 
therefore you live'. 

21.19 Such an argument as that which says: 'it is day; not: 
it is not day; therefore it is light' has potentially a single 
premiss. 

Further, apparently numberless, divisions of arguments are 
obscure in our sources. Diogenes speaks of 'possible, impossible, 
necessary and not necessary' arguments (21.20). Sextus has a 
division into demonstrable and indemonstrable arguments, the 
last-named being either simple or compound, and the compound 
being reducible to the simple (which makes them demonstrable), 
(21.21). The whole account is so vague that we are not in a position 
to grasp the meaning of this division. But in Diogenes we find a 
consistent doctrine of these same 'indemonstrable' arguments; 
they are simply the axioms of the Stoic propositional logic, and we 
consider them in the next chapter. 



D. SCHEMES OF INFERENCE 

The Stoics made clear distinction between a logical rule and an 
instance of it, i.e. between the moods (Tpo7roi) of an argument 
and the argument itself - a distinction which Aristotle applied in 
practice, but without a theoretic knowledge of it. 

21.22 These are some of the arguments. But their moods or 
schemata (ox^octoc) in which they are formed are as follows: 
of the first indemonstrable: 'if the first, then (the) second; the 
first; therefore the second'; of the second: 'if the first, (then) 
the second; not the second; therefore not the first'; of the 
third : 'not : the first and the second ; the first; therefore not the 
second'. 

We have similar schemata for other arguments as well (21.23 . 
even for some of the not indemonstrable (cf. 22.17). It is striking 
that only numerical words occur in them as variables. One might 
conjecture that this was so in Aristotle too, since in Greek the 
letters of the alphabet could function as numerals; but the fact 

125 



THE GREEK VARIETY OF LOGIC 

that Aristotle did not only use the early letters of the alphabet, 
but often IT, P, and 2 as well, seems to exclude this. 

Along with these homogeneous formulae the Stoics also had 
'mixed' half-arguments, half-schemata. They were called 'argument- 
schemata' (Xoy6Tpo7iot,). 

21.24 An argument-schema consists of both, e.g. 'if Plato 
lives, Plato breathes; the first; therefore the second'. The 
argument-schema was introduced in order not to have to 
have a long sub-premiss in long formulae, so as to gain the 
conclusion, but as short as possible: 'the first; therefore the 
second'. 

Another example is this: 

21.25 If the sweat flows through the surface, there are 
intelligible pores; the first; therefore the second. 



§22. AXIOM ATI ZAT I ON. COMPOUND ARGUMENTS 

The Stoic propositional logic seems to have been thoroughly 
axiomatized, distinction even being made between axioms and 
rules of inference. 

A. THE INDEMONSTRABLES 

The tradition is obscure about the axioms (22.01 ; et vid. supra 
21.21). We here give the definition of the indemonstrables according 
to Diogenes, their description, with examples, from Sextus. 

22.02 There are also some indemonstrables (av<x7c68eixToi) 
which need no demonstration, by means of which every (other) 
argument is woven; they are five in number according to 
Chrysippus, though other according to others. They are 
assumed in conclusives, syllogisms and hypotheticals (Tp07ux£>v) 

22.03 The indemonstrables are those of which the Stoics 
say that they need no proof to be maintained. . . . They 
envisage many indemonstrables, but especially five, from 
which it seems all others can be deduced. 

This is no less than an assertion of the completeness of the system : 
whether it is correct we cannot tell, since we do not know the 
metatheorems and have only a few of the derivative arguments. 

126 



MEGARIAN-STOIG LOGIC 

22.04 The first (indemonstrable) from a connected (pro- 
position) and its antecedent yields its consequent, e.g. 'if 
it is day, it is light; it is day; therefore it is light'; 

22.05 the second from a connected (proposition) and the 
contradictory opposite (avTixst^evou) of its consequent yields 
the contradictory opposite of its antecedent, e.g. 'if it is 
day, it is light; it is not light; therefore it is not day' ; 

22.06 the third from the negation (obccxpaTixoO) of a con- 
junction together with one of its components, yields the 
contradictory opposite of the other, e.g. 'not: it is day and 
it is night; it is day; therefore it is not night' ; 

22.07 the fourth, from a (complete) disjunctive (proposi- 
tion) together with one of the (propositions) disjoined (e7te£evy- 
(jtivoov) in it, yields the contradictory opposite of the other, e.g. 
'either it is day or it is night; it is day; therefore it is not 
night' ; 

22.08 the fifth from a (complete) disjunctive (proposition) 
together with the contradictory opposite of one of the dis- 
joined (propositions) yields the other, e.g. 'either it is day or it 
is night; it is not night; therefore it is day'. 

Other less reliable sources speak of two further indemonstrables, 
the sixth and seventh (22.09). 

B. METATHEOREMS 

The reduction of demonstrable arguments to indemonstrable, was 
effected in Stoic logic by means of certain metalogical rules. One 
name for such was Os(jia, but it seems that the expression OscopTjpia 
was also used (22.10). We shall call them 'metatheorems', in accord- 
ance with modern usage. A text of Galen shows that there were 
at least four of them (22.11), but only the first and third are stated 
explicitly. 

22.12 There is also another proof common to all syllogisms, 
even the indemonstrable, called '(reduction) to the impossible' 
and by the Stoics termed 'first metatheorem' (constitutio) or 
'first exposition' (expos Hum). It is formulated thus: 'If some 
third is deduced from two, one of the two together with the 
opposite of the conclusion yields the opposite of the other.' 

This is the rule for reduction to the impossible (16.33), already 
stated by Aristotle in another form. 

22.13 The essentials of the so-called third metatheorem 
(SepuxTCK;) look like this : if some third is deduced from two and 

127 



THE GREEK VARIETY OF LOGIC 

one (of the two) can be deduced syllogistically from others, 
the third is yielded by the rest and those others. 

This metatheorem is what in fact underlies the aristotelian 
'direct reduction' of syllogisms, and can be formulated: 

22.131 If r follows from p and q, and p from s, then r 
follows from q and s (cf. 14.141). 

The following is given by Alexander as the 'synthetic theorem' 
(cjuvOstixov 0s:copY][jLa) : 

22.14 If some (third) is deduced from some (premisses), and 
if the deduced (third) together with one or more (fourth) 
yields some (fifth), then this (fifth) is deduced also from those 
(premisses) from which this (third) is deduced. 

The rule being stated is this: 

22.141 If r follows from p and g, and t from r and s, 
then i follows from p, q and s; 

or, if one represents the premisses with a single variable : 

22.142 If q follows from p, and s from q and r, then s 
follows also from p and r. 

Sextus cites a similar but seemingly different metatheorem: 

22.15 It should be known that the following dialectical 
theorem (OscopY^oc) has been handed down for the analysis of 
syllogisms : 'if we have premisses to yield a conclusion, then we 
have this conclusion too potentially among these (premisses), 
even if it is not explicitly (xoct* excpopav) stated. 

We have two detailed examples of the apllication of this meta- 
theorem, which belong to the highest development of Stoic logic. 

C. DERIVATION OF COMPOUND ARGUMENTS 

22.16 Of the not-simple (arguments) some consist of 
homogeneous, others of not homogeneous (arguments). Of 
not homogeneous, those which are compounded of two first 
indemonstrables (22.04), or of two second (22.05). Of not 
homogeneous, those which (are compounded) of second and 
third* (22.06), or in general of such (dissimilars). An example of 

* Reading xal Tptxou in the lacuna with Kochalsky. 

128 



MEGARIAN-STOIC LOGIC 

those consisting of homogeneous (arguments) is the following: 
'if it is day*, then if it is day it is light; it is day; therefore it 
is light'. . . . For we have here two premisses, (1) the connected 
proposition: 'if it is day* *, then if it is day it is light', which 
begins with the simple proposition 'it is day' and ends with 
the not simple, connected proposition 'if it is day it is light'; 
and (2) the antecedent in this (first premiss:) 'it is day'. If 
by means of the first indemonstrable we infer from those the 
consequent of the connected (proposition, viz.) 'if it is day it 
is light', then we have this inferred (proposition) potentially in 
the argument, even though not explicitly stated. Putting this 
now together with the minor premiss of the main argument, 
viz. 'it is day', we infer by means of the first indemonstrable: 
'it is light', which was the conclusion of the main argument. 

22.17 That is what the arguments are like which are com- 
pounded from homogeneous (indemonstrables). Among the 
not homogeneous is that propounded by Ainesidemus about 
the sign, which runs thus: 'if all phenomena appear similarly 
to those who are similarly disposed, and signs are phenomena, 
then signs appear similarly to all those who are similarly 
disposed; signs do not appear similarly to all those who are 
similarly disposed; phenomena appear similarly to all those 
who are similarly disposed ; therefore signs are not phenomena'. 
This argument is compounded of second and third indemon- 
strables, as we can find out by analysis. This will be clearer if 
we put the process in the form of the schema of inference: 
'if the first and second, then the third ; not the third ; the first; 
therefore not the second'. For we have here a connected (propo- 
sition) in which the conjunction of the first and second forms 
the antecedent, and the third the consequent, together with 
the contradictory opposite of the consequent, viz. 'not the 
third'. Hence we infer by means of the second indemonstrable 
the contradictory opposite of the antecedent, viz. 'therefore 
not: the first and the second'. But this is potentially contained 
in the argument, as we have it in the premisses which yield it, 
though not verbally expressed. Putting it * * * together with the 
other premiss, the first, we infer the conclusion (of the main 
argument), 'not the second', by means of the third indemon- 
strable. 

* Adding with Kochalsky: si yjfiipa saxtv. 
* * Completing the text as before. 
* * * reading 07rsp instead of a7rep, with Kochalsky. 

129 



THE GREEK VARIETY OF LOGIC 
D. FURTHER DERIVED ARGUMENTS 

According to Cicero (22.18) the Stoics derived 'innumerable' 
arguments in similar ways. 

22.19 The said (Chrysippus) says that it (the dog) often* 
applies the fifth indemonstrable, when on coming to the 
meeting of three roads it sniffs at two down which the game 
has not gone and immediately rushes down the third without 
sniffing at it. The sage says in fact that it virtually infers: 
the game has gone down this, or this, or that; neither this nor 
this; therefore that. 

22.21 If two connected (propositions) end in contradictorily 
opposed (consequents) - this theorem is called (the theorem) 
'from two connecteds' (Tpo7cixc5v) - the (common) antecedent 
of the two is refuted .... This argument is formed according to 
the schema of inference : 'if the first, the second ; if the first * *, 
not the second; therefore not the first'. The Stoics give it 
material expression (i.e. by a substitution) when they say 
that from the (proposition) 'if you know that you are dead 
(you are dead if you know that you are dead) you are not dead' 
there follows this other: 'therefore you do not know that you 
are dead'. 

22.22 Some argue in this way: 'if there are signs, there are 
signs; if there are not signs, there are signs; there are either no 
signs or there are signs; therefore there are signs'. 



§ 23. THE LIAR 

The Stoics and above all the Megarians devoted much attention 
to fallacies. Some of the ones they considered derive from the problem 
of the continuum and belong to mathematics in the narrower sense 
of that word ; the rest are mostly rather trifles than serious logical 
problems (23.01). But one of their fallacies, 'the Liar' (^£uSo[X£vo<;) 
has very considerable logical interest and has been deeply studied 
by logicians for centuries, in antiquity, the middle ages, and the 
20th century. The Liar is the first genuine semantic fallacy known to 
us. 



* Sia tcXeiovcov: this could also mean 'the (argument) from the more'; but I 
follow Mates (22.20) since (1) we know of no such indemonstrable, and (2) the 
argument is reducible to the simple fifth indemonstrable. 
* * omitting ou with Koetschau. 

130 



MEGARIAN-STOIC LOGIC 
A. HISTORY 

In St. Paul is to be found the following notable text: 

23.02 One of themselves, a spokesman of their own, has 
told us : The men of Crete were ever liars, venomous creatures, 
all hungry belly. 

According to various sources (23.03) this spokesman was Epimeni- 
des, a Greek sage living at the beginning of the 6th century b.c. 
Hence the Liar is often called after him, but wrongly, for Epimenides 
was clearly not worrying about a logical paradox. Plato, too, who con- 
sidered similar problems in the Euthydemus (ca. 387 B.C.; 23.04; did 
not know the Liar. But Aristotle has it in the Sophistic Refutations, 
about 330 b.c. (23.05). Now that is just the period when Eubulides 
was flourishing, to whom Diogenes Laertius explicitly ascribes the 
discovery of the Liar (18.02). After that, Theophrastus wrote three 
books on the subject (23.06), Chrysippus many more, perhaps 
twenty-eight (23.07). How much people took the problem to heart 
at that time can be seen from the fact that one logician, Philetas of 
Cos (ca. 340-285 b.c), died because of it: 

23.08 Traveller, I am Philetas; the argument called the 
Liar and deep cogitations by night, brought me to death. 

B. FORMULATION 

In spite of this interest and the extensive literature about the 
Liar, we no longer possess Eubulides's formulation of the antinomy, 
and the versions that have come down to us are so various that it 
is impossible to determine whether a single formula underlies them 
all, and which of the surviving ones has been considered by com- 
petent logicians. Here we can only give a simple list of the most 
important, collected by A. Riistow (23.09). They seem to fall into 
four groups. 

I. 

23.10 If you say that you lie, and in this say true, do you 
lie or speak the truth ? 

23.11 If I lie and say that I lie, do I lie or speak the truth? 

II 

23.12 If you say that you lie, and say true, you lie; but 
you say that you lie, and you speak the truth; therefore you 
lie. 

23.13 If you lie and in that say true, you lie. 

131 



THE GREEK VARIETY OF LOGIC 
III. 

23.14 I say that I lie, and (in so saying) lie; therefore I 
speak the truth. 

23.15 Lying, I utter the true speech, that I lie. 

IV. 

23.16 If it is true, it is false ; if it is false, it is true. 

23.17 Whoso says 'I lie', lies and speaks the truth at the 
same time. 

The relation of the four groups to one another is as follows : the 
texts of the first group simply posit the question: is the Liar true or 
false ? Those of the second conclude that it is true, of the third that 
it is false. The texts of the fourth group draw both conclusions 
together; the proposition is both true and false. 

C. EFFORTS AT SOLUTION 

Aristotle deals with the Liar summarily in that part of his 
Sophistic Refutations in which he discusses fallacies dependent on 
what is said 'absolutely and in a particular respect': 

23.18 The argument is similar, also, as regards the problem 
whether the same man can at the same time say what is both 
false and true: but it appears to be a troublesome question 
because it is not easy to see in which of the two connections 
the word 'absolutely' is to be rendered - with 'true' or with 
'false'. There is, however, nothing to prevent it from being 
false absolutely, though true in some particular respect (nfi) or 
relation (tiv6<;), i.e. being true in some things though not 
true absolutely. 

It has been said (23.19) that the difficulty is here 'quite unresolved, 
and indeed unnoticed', and indeed Aristotle has not solved our 
antinomy nor understood its import. Yet, as is so often the case 
with this pastmaster, he reveals a penetrating insight into the 
principle of the medieval and modern solutions - the necessity of 
distinguishing different aspects, levels as we now say, in the Liar. 
Worth noting too, is Aristotle's standpoint of firm conviction that 
a solution is discoverable. This conviction has remained the motive 
power of logic in this difficult field. 

The solution of Chrysippus has reached us in a very fragmentary 
papyrus, written moreover in difficult language. Its essential, and 
most legible, part is as follows* : 

* Thanks are due to Prof O. Gigon for help with this text. 

132 



MEGARIAN-STOIC LOGIC 

23.20 The (fallacy) about the truth-speaker and similar 
ones are to be . . . (solved in a similar way). One should not say 
that they say true and (also) false; nor should one conjecture 
in another way, that the same (statement) is expressive of true 
and false simultaneously, but that they have no meaning at all. 
And he rejects the afore-mentioned proposition and also the 
proposition that one can say true and false simultaneously and 
that in all such (matters) the sentence is sometimes simple, 
sometimes expressive of more. 

The most important words in this text are cn)[i.ar.vofiivoi> ziXzac, 
(X7ro7rAavc5vTai, translated '(that) they have no meaning at all'. 
The Greek phrase is ambiguous as between (1) that whoever states 
the Liar attributes a false assertion to the proposition, and (2) that 
he says something which has no meaning at all. The fragmentary 
context seems to indicate the second interpretation as the correct 
one, but it is impossible to be certain of this. If it is correct, Ghrysip- 
pus's solution is that the Liar is no proposition but a senseless 
utterance, which would be a view of the highest importance. The 
Aristotelian attempt to solve it is definitely rejected in this text. 



133 



IV. THE CLOSE OF ANTIQUITY 



§24. PERIOD OF COMMENTARIES 
AND HANDBOOKS' 

A. CHARACTERISTICS AND HISTORICAL SURVEY 

With the end of the old Stoa there begins a period into which 
hardly any research has been done. However, on the basis of the 
few details known to us we may suppose with great probability that 
the formal logic of this period was of the following kind : 

1. The period is not a creative one. No new problems or original 
methods such as those developed by Aristotle and the Megarian- 
Stoic school are to be found. 

2. Yet, apparently right up to the fall of the Roman empire, 
individual scholarly works appeared. Some earlier methods were 
improved, the material was systematized and sometimes developed. 
There were even not wanting genuinely gifted logicians, among the 
best of whom was Alexander of Aphrodisias. 

3. The logical literature consisted chiefly of two kinds of work: 
big commentaries, mainly on Aristotle, and handbooks. 

4. As to their content, we discern mostly a syncretizing tendency 
in the sense that Aristotelian and Stoic-Megarian elements are 
mingled, Stoic methods and formulations being applied to Aristo- 
telian ideas. 

Lack of monographs makes it impossible to survey the state of the 
logical problematic during the period, and we limit ourselves to the 
choice of some particular doctrines so far found in the mass of 
commentaries and handbooks. But first some of the most important 
thinkers must be named. 

The first well-known logicians of this period are Galen and the 
less notable Apuleius of Madaura whose handbooks have survived; 
the former is the subject of the only monograph on the period 
(24.01). In the 3rd century a.d. we find Alexander of Aphrodisias, 
already mentioned, one of the best commentators on the whole 
Aristotelian logic, and unlike Galen and Apuleius a fairly pure 
Aristotelian. Porphyry of Tyre lived about the same time, and 
composed an Introduction (ziGOLycxtyri) to the Aristotelian categories. 
In it he systematized the doctrine of the predicables (11.06ff.), 
giving a five-fold enumeration: genus, specific difference, species, 
property and accident (24.02). This work was to be basic in the 
Middle Ages. Later logicians include Iamblichus of Chalcis, not to 
be taken very seriously, Themistius (both these in the 4th century 
a.d.), Ammonius Hermeae (5th century), Martianus Capella, author 

134 



TREE OF PORPHYRY 

of a handbook which formed an important link between ancient 
and later logic (5th century), AmmoniuB the Peripatetic, Simplicius 
(6th century), who was another of the better commentators on 
Aristotle, and finally Philoponus (7th century), but these have 
little importance so far as we can judge. On the other hand the last 
Roman logician, Boethius (ca. 480-524) is of fairly considerable 
importance both because his works became a prime source for the 
Scholastics and also because he transmits doctrines and methods 
not mentioned elsewhere, though he himself was only a moderate 
logician. With his execution the West enters on a long period without 
any logic worth speaking of. 



B. THE TREE OF PORPHYRY 

Of the commentators' discoveries the 'tree of Porphyry' has 
certainly achieved the greatest fame. While it can be regarded as 
only a compendium of Aristotelian doctrines it has great importance 
as comprising (1) a system of classification, which was not to the 
fore in Aristotle's thought (11.13), and (2) an extensional view of 
terms. First we give the text: 

24.03 Let what is said in one category now be explained. 
Substance (ouctioc) is itself a genus, under this is body, and under 
body is living (s^uxov) body, under which is animal. Under 
animal is rational (Xoyixov) animal, under which is man. Under 
man are Socrates and Plato and individual (xorra uipo<;) men. 
But of these, substance is the most generic and that which is 
genus alone ; man is the most specific and that which is species 
alone. Body is a species of substance, a genus of living body. 

The following text shows how thoroughly extensional a view is 
being taken: 

24.04 (Genus and species) differ in that genus contains 
(■7T£ptix £L ) its species, the species are contained in but do not 
contain their genus. For the genus is predicated of more things 
than the species. 

This conception is carried so far that one can here properly speak 
of a beginning of calculus of classes. At the same time Porphyry 
makes a distinction which corresponds fairly closely to the modern 
distinction between extension and intension (36.10, 45.03) - or. 
again, between simple and personal supposition (27.15). For among 
a number of definitions of the predicables, he has : 

135 



CLOSE OF ANTIQUITY 

24.05 The philosophers . . . define, saying that genus is what 
is predicated essentially (ev tw tl eoti) of several things 
differing in species. 

24.06 The genus differs from the difference and the common 
accidents in that, while the difference and the common acci- 
dents are predicated of several things differing in species, they 
are not predicated essentially but as qualifying (ev tco o7rot6v tl 
e<mv). For when we ask what it is of which these are predicated, 
we answer with the genus; but we do not answer with the 
differences or accidents. For they are not predicated essentially 
of the subject but rather as qualifying it. For on being asked 
of what quality man is, we say that he is rational, and to the 
question of what quality crow is, we answer that it is black. 
But rational is a difference, and black an accident. But when 
we are asked what man is, we answer that he is animal, animal 
being a genus of man. 

C. EXTENSION OF LOGICAL TECHNIQUE 

Among the most important achievements of this period are two 
devices which so far as we know were unknown to Aristotle and the 
Stoics, viz. (1) identification of variables, (2) substitution of sen- 
tential forms for variables. 

1. Alexander of Aphrodisias 

The first is to be found in Alexander in a new proof of the con- 
vertibility of universal negative sentences : 

24.07 If someone were to say that the universal negative 
(premiss) does not convert, (suppose) A belongs to no B; if 
(this premiss) does not convert, B belongs to some A; there 
results in the first figure (the conclusion that) to some A A 
does not belong, which is absurd. 

Alexander here makes use of the fourth syllogism of the first 
figure (Ferio: (13.06), which in Aristotle's presentation runs: 'if A 
belongs to no B, but B to some C, then to some C A cannot belong'. 
He identifies C with A - i.e. substitutes one variable for the other, 
and obtains: 'if A belongs to no B, but B to some A, then to some A 
A does not belong'. That is the novelty of the process. 

This is consonant with Alexander's clear insight into the nature 
of laws of formal logic. He seems to have been the first to make 
explicit the distinction between form and matter, and at the same 
time to have come close to an explicit determination of the notion 
of a variable. 

136 



IMPLICATION 

24.08 He (Aristotle) introduces the use of letters in order 
to show us that the conclusions are not produced in virtue 
of the matter but in virtue of such and such a form 
(oXWol) and composition and the mood of the premisses; the 
syllogism concludes . . . not because of the matter, but because 
the formula (au^uyta) is as it is. The letters show that the 
conclusion is of such a kind universally and always and for 
every choice (of material). 

2. Boethius 

A further development of the technique of formal logic is to be 
found in Boethius. He is evidently aiming at the formulation of a 
rule of substitution for propositional variables ; this is not given in the 
form of such a rule, but in a description of the structure of formulae. 
Again we have a fairly clear distinction between form and matter in 
a proposition, a distinction which was to play a great part in later 
history : 

24.09 We shall now show the likenesses and differences 
between simple propositions and compound hypothetical ones. 
For when the (hypothetical propositions) which consist of 
simple ones are compared with those which are compounded of 
two hypotheticals, (one sees that) the sequence (in both cases) 
is the same and the relation (of the parts to one another) 
remains, only the terms are doubled. Since the places which are 
occupied by simple propositions in those hypotheticals consist- 
ing of simple propositions, are occupied in hypotheticals consist- 
ing of hypotheticals by those conditions in virtue of which those 
(component) propositions are said to be joined and linked 
together. For in the proposition which says: 'if A is, B is', 
and in that which says: 'if, if A is, B is, (then) if C is, D is' 
the place occupied in that consisting of two simple propositions 
by that which is first: 'if A is', in the proposition consisting of 
two hypotheticals is occupied by that which (there) is first: 'if, 
if A is, B is'. 

If we remember the Stoic distinction between argument and mood 
(21.22) the last two texts do not seem very original; but they are the 
first in which an explicit statement of the distinction is found. 

D. FRESH DIVISION OF IMPLICATION 
It is Boethius again who gives a fresh division of implication : 

24.10 Every hypothetical proposition is formed either by 
connection (connexionem) ... or by disjunction. . . . But since 

137 



CLOSE OF ANTIQUITY 

it has been said that the same thing is signified by the connec- 
tives (conjunctione) 'si' and 'cum' when they are put in hypo- 
thetical sentences, conditionals can be formed in two ways: 
accidentally, or so as to have some natural consequence. 
Accidentally in this way, as when we say: 'when fire is hot, 
the sky is round'. For the sky is round not because fire is hot, 
but the sentence means that at what time fire is hot, the 
sky is round. But there are others which have within them 
a natural consequence, . . . e.g. we might say: 'when man is, 
animal is'. 

There is here, as often elsewhere, a certain obscurity in Boethius's 
thought (24.11). Apart from that, his division of implication is 
something of a backward step in comparison with the Stoic dis- 
cussions of the subject (vide supra 20.05 ff.). Yet the text just cited is 
important for our history, being an evident starting-point for scholas- 
tic speculations about implication. 

Hence also we mention the following details of Boethius's doc- 
trine about propositional functors. He often seems to use l sV (24.12) 
as a symbol of equivalence (cf. 20.20 ff.). The sense of the expression 
k auV is ambiguous. On the one hand, we find - and for the first 
time - a definition in the sense of non-exclusive alternation (logical 
sum : cf. 20.17, 30.18, 40.11, 41.18) : 

24.13 The disjunctive proposition which says (proponit) : 
'either A is not or B is not' is true (fit) of those things which 
can in no way co-exist, since it is also not necessary that 
either one of them should exist; it is equivalent to that com- 
pound proposition in which it is said : 'if A is, B is not'. ... In 
this proposition only two combinations yield (valid) syllogisms. 
For, if A is, B will not be, and if B is, A will not be. . . . For 
if it is said: 'either A is not or B is not', it is said: 'if A is, B 
will not be', and 'if B is, A will not be'. 

First we have here Sheffer's functor ('not p or not q'-, 43.43); 
secondly this text contains an exact definition of the logical sum. 
The essential idea can be formulated : 

24.131 Not p or not q if and only if : if p then not q. 

Putting therein 'not-p' for 'p' and 'not-g' for l q\ we get by the 
principle of double negation: 

24.132 p or q if and only if: if not-p then q. 
138 



HYPOTHETICAL SYLLOGISMS 

On the other hand, Boethius defines in analogous fashion his 

l sV in the sense of equivalence by means of the same 'auV - which 
therefore and in this case has the sense of aegaied equivalence 
(p or q but not both, and necessarily one of the two) (24.14). 

It is also worth remarking that Boethius regularly uses the 
principle of double negation and a law analogous to 24.21. 

E, BOETHIUS'S HYPOTHETICAL SYLLOGISMS 

We here give the list of Boethius's hypothetical syllogisms. They 
seem to be the final result of Stoic logic, if understood as laws of the 
logic of propositions. Our supposition that Boethius aspired to a rule 
of substitution for propositional variables (cf. 24.09), requires them to 
be so understood. They would be the final result of Stoic logic in the 
sense that they are practically the only part of this logic that was 
preserved by Boethius for the Middle Ages. 

24.15 If A is, B is; but A is; therefore B is. 

24.16 If A is, B is; but B is not; therefore A is not. 

24.17 If A is, B is, and if B is, C must be; but then: if A is, 
C must be. 

24.18 If A is, B is, and if B is, C too must be; but C is not; 
therefore A is not. 

24.19 If A is, B is ; but if A is not, C is ; I say therefore that 
if B is not, C is. 

24.20 If A is, B is not; if A is not, C is not; I say therefore 
that if B is, C is not. 

24.21 If B is, A is; if C is not, A is not; on this supposition 
I say that if B is, it is necessary that C is not. 

24.22 If B is, A is; if C is not, A is not; I say therefore: if 
B is is, C will be. 

24.23 If one says : 'either A is or B is', (then) if A is, B \x\\\ 
not be; and if A is not, B will be; and if B is not, A will be; 
and if B* is, A will not be. 

24.25 The (proposition) that says: 'either A is not or B is 
not', certainly means this, that if A is, B cannot be. 

Boethius developes these syllogisms by substituting a conditional 
proposition for one or both variables (cf. 24.09) ; in so doing he treats 
the negation of a conditional as the conjunction of the antecedent 
with the negation of the consequent, according to the law. which is 
not expressly formulated: 

24.251 Not: if p, then q, if and only if: p and not-g. 

* omitting non with van den Driessche (24.24). 

139 



CLOSE OF ANTIQUITY 

Finally he applies the law of double negation (cf. 20.041), thus 
gaining eighteen more syllogisms (24.26). 



F. ALTERATIONS AND DEVELOPMENT OF THE CATEGORICAL 

SYLLOGISTIC 

24.27 But Ariston the Alexandrian and some of the later 
Peripatetics further introduce five more moods (formed from 
those) with a universal conclusion: three in the first figure, 
two in the second figure, which yield particular conclusions. 
(But) it is extremely foolish to conclude to less from that to 
which more is due. 

This text is not very clear. But its difficulty is somewhat lessened 
if we suppose that a combination of two Aristotelian rules is envi- 
saged: (1) that allowing a universal conclusion to be weakened to 
the corresponding particular (13.23), (2) that yielding a further 
conclusion by conversion of the one first obtained. Then the follow- 
ing would be the moods intended : 

24.271 A to all B; B to all C; therefore A to some C 
(Barbari). 

24.272 A to no B; B to all C; therefore to some C, A not 
(Celaront). 

24.273 A to all B; B to some C ; therefore C to some A 
(Dabiiis). 

ZZk.Zlb B to no A; B to all C; therefore to some C, A not 
(Cesaro). 

24.275 B to all A; B to no C ; therefore to some C, A not 
(Camestrop). 

Beyond these, Galen transmits a further mood of this kind in the 
third figure (24.28): 

24.281 A to all B; C to all B; therefore C to some A 
(Daraptis). 

These formulae all have a Stoic rather than an Aristotelian form. 
In fact from Apuleius on, such alteration of the old laws into rules 
is more or less standard practice, especially in Boethius. 

A further precision given to the Aristotelian syllogistic is in the 
famous logical square. This figure is first found in Apuleius again. It 
looks like this: 



140 



FOURTH FIGURE 



24.29 





Contrariae 


vd 


incongruae 






c 

iS 


universal affirmative 


W 


universal negal ive 


1 


all pleasure is good 


no pleasure is good 






some pleasure is good 


some pleasure is not good 


particular affirmative 


particular negative 




Subconlrariae 


vel 


sub pares 





G. THE SUPPOSEDLY FOURTH FIGURE 

In an anonymous fragment, belonging possibly to the 6th cen- 
tury, we read : 

24.30 Theophrastus and Eudemus also added other formu- 
lae to those of Aristotle in the first figure . . . many moderns 
have thought to form the fourth figure therefrom, citing 
Galen as the author of this intention. 

But this allegedly 'Galenic' figure is not to be found in him. On 
the contrary he plainly states that there are only three figures : 

24.31 These syllogisms are called, as I have said, categorical ; 
they cannot be formed in more than the three figures men- 
tioned, nor in another number in each (of these figures); this 
has been shown in the treatises On Demonstration. 

J. Lukasiewicz was able to explain by means of another anony- 
mous fragment how nevertheless the discovery of the fourth figure 
could be credited to Galen (24.32). This fragment is not without 
historical interest even apart from this question: 

24.33 Of the categorical (syllogism) there are two kinds; 
the simple and the compound. Of the simple syllogism there 
are three kinds: the first, the second, and the third figure. Of 
the compound syllogism there are four kinds: the first, the 
second, the third, and the fourth figure. For Aristotle says 
that there are only three figures, because he looks at the simple 
syllogisms, consisting of three terms. Galen, however, says 
in his Apodeidic that there are four figures, because he looks at 
the compound syllogisms consisting of four terms, as he has 
found many such syllogisms in Plato's dialogues. 

141 



CLOSE OF ANTIQUITY 

24.34 The categorical syllogism 

simple, as ( in) Aristotle compound, as (in) Galen 

Figure 1,2,3 

Compound figure 
1 to 1, 1 to 2, 1 to 3, 2 to 2, 2 to 1, 2 to 3, 3 to 3, 3 to 1, 3 to 2. 



1 to 1 

1 



Compound figure 

syllogistic : 

1 to 2 1 to 3 

2 3 



2 to 3 
4 



2 to 2 3 to 3 
since no syllogism 
arises from two 
negatives or two 
particulars. 

1 to 1, as in the Alcibiades 



unsyllogistic : 

2 to 1 3 to 1 3 to 2 
2 3 4 




useful 



The numerals denote the successive figures, and the author 
means that a valid compound syllogism can be formed in four 
different ways, viz. when of the two simple syllogisms from which it 
is composed 

(1) both are in the first figure, 

(2) the first is in the first, the second in the second figure. 

(3) the first is in the first, the second in the third, 

(4) the first is in the second, the second in the third. 

Those are the four figures. So there is no question of a fourth figure of 
simple syllogism, which was only ascribed to Galen by a misunder- 
standing. Yet the unknown scholiast (24.30), in falling a victim to 
this misunderstanding at least made the principle of the fourth 
figure another interpretation of the indirect moods of Theophrastus 
(17.10). 

142 



PONS ASINORUM 



H. PONS ASINORUM 



Here we should introduce a scheme which was to become famous 
in the Middle Ages as the pons asinorum or 'asses' bridge' 'I'l.Xil). . 
It is to be found in Philoponus*, and is an elaboration of the 
Aristotelian doctrine of the inuentio medii (14.29;. Although it 
belongs to methodology rather than logic, it is relevant to the latter 
also. The scheme seems typical of the way in which the commentators 
developed the syllogistic. In Philiponus the lines are captioned in the 
figure itself. For graphical reasons we put these comments after- 
wards and refer to them by numbers. 



24.35** 



GOOD 

A 



What follows on the good 

helpful, eligible, 

to be pursued, 

suitable, desirable, 

profitable, 

expedient. 



What is alien to the good: -4 
imperfect, to be 
fled from, harmful, 
bad, ruinous, alien, 
unprofitable. 

What the good 
follows upon: 
happiness, natural 
well-being, final 
cause, perfect, 
virtuous life. 




PLEASURE 
E 
What follows on 
pleasure: 

movement, natural 
activity, unimpeded 
life, object 
of natural desire, 
undisturbed, 
eligible. 

What is alien to 
pleasure: 
disease, labour, 
fear, need, 
unnatural movement. 

What pleasure 
follows upon: 
health, good 
repute, virtuous 
life, plenty, 
good children, freedom 
from pain, comfort. 



1) Unsyllogistic, because of concluding in the second figure 
from two universal affirmative (premisses). 

2) Universal negative (conclusion) in the first and second 
(figures) by two conversions. 

3) Particular affirmative (conclusion in the first and third) 
figures by conversion of the conclusion. 

4) Universal negative (conclusion) in the first and second 
figure. 



* Thanks are due to Prof. L. Minio-Paluello for pointing out this passage. 
* * For typographical reason the words in the figure are set in small type, 
though they belong to the quotation. 



143 



CLOSE OF ANTIQUITY 

5) Universal affirmative (conclusion) in the first figure. 

6) Unsyllogistic, from two universal negatives. 

7) Particular negative (conclusion) in the third and first 
(figure) through conversion of the minor (premiss). 

8) Unsyllogistic, since the particular does not convert, and in 
the first (figure) because (the syllogism) has a negative minor 
(premiss). 

9) Particular affirmative (conclusion) in the third and first 
figures by conversion of the minor premiss. 

I. ANTICIPATION OF THE LOGIC OF RELATIONS 

Finally we shall speak of a detail which had no influence on the 
later development of logic, but which yet may be reckoned an 
ingenious anticipation of the logic of relations. Galen, dividing 
syllogisms in his Introduction, distinguishes first between categorical 
and hypothetical syllogisms, thus separating term- and class-logic; 
he then adds a further class: 

24.36 There is still a further, third class of syllogisms, 
useful for demonstration, which I characterize as based on 
relation. Aristotelians claim that they are counted as cate- 
goricals. They are not a little in use among the Sceptics, 
Arithmeticians and experts in calculation in certain arguments 
of this kind: 'Theon possesses twice what Dion possesses; but 
Philo too possesses twice what Theon possesses; therefore 
Philo possesses four times what Dion possesses.' 

This is in fact a substitution in a law of the logic of relations, and 
it is remarkable that Galen divides his logic just as Whitehead and 
Russell were to do in the 20th century. The content of his logic of 
relations is of course very poor, and he thinks that such laws are 
reducible to categorical syllogisms (24.37), which is a regress from 
the position of Aristotle. 

SUMMARY 

To summarize the results of post-Aristotelian antiquity we can 
say: 

1. Propositional logic was then created. Some theorems of this 
kind were already known to Aristotle, sometimes even stated with 
propositional variables: but these were rather obiter dicta than 
systematically presented. In the Stoics on the other hand we meet 
systematic theory developed for its own sake. 

2. This system is based on a fairly well worked out semantics, 
and it was expressly stated in the Stoic school that it was concerned 

144 



SUMMARY 

neither with words nor psychic images, hut with objective meanings, 
the lecla. We have therefore to thank them for a fundamental 
thesis which was to play a great part in the history of logic. 

3. Megarian-Stoic logic contained an astonishingly exact analysis 
of proposition-forming functors: we find correctly formed truth-tables 
and a more intricate discussion of the meaning of implication than 
we seem yet to have attained in the 20th century. 

4. In this period the method is formalistic. Unambiguous 
correlation of verbal forms to lecla being presupposed, attention is 
exclusively directed to the syntactical structure of expressions. 
The application of this method and the logical subtlety shown by the 
Stoics must be deemed quite exemplary. 

5. This formalism is accompanied by a significant extension of 
logical technique, shown in the clear distinction between propositional 
functions and propositions themselves, the method of identification 
of variables, and the application of the rule permitting substi- 
tution of propositional functions for propositional variables. 

6. Propositional logic is axiomatized, and a clear distinction 
drawn between laws and melalheorems. 

7. Finally we have to thank the Megarian school for propound- 
ing the first important logical antinomy - the Liar - which for 
centuries remained one of the chief problems of formal logic, and is 
so even today. So without exaggeration one can say that the achieve- 
ments of this period make up antiquity's second basic contribution 
to formal logic. 



145 



PART III 



The Scholastic Variety of Logic 



§25. INTRODUCTION TO SCHOLASTIC LOGIC 



A. STATE OF RESEARCH 

At the present time much less is known about the history of 
scholastic than of ancient logic. The reason is that when Scholasti- 
cism ceased to be disparaged at the end of the nineteenth century, 
there was at first little revival of interest in its formal logic. This 
lack of interest is shown in the fact that of more than ten thousand 
titles of recent literature on Thomas Aquinas (up to 1953), very 
few concern his formal logic. There are indeed earlier works treating 
of questions of the literary history of scholastic logic - Grabmann 
having done most to find and publish texts -, but the investigation 
of their logical content only began with Lukasiewicz's paper of 
1934 (25.01), pioneering in this field too. Under his influence some 
notable medievalists, e.g. besides Grabmann, K. Michalski, applied 
themselves to logical problems, and from his school there came the 
first work, well and systematically prepared, on medieval logic, the 
paper on the propositional logic of Ockham by J. Salamucha (1935) 
(25.02). A number of texts and treatises followed, those of Ph. Boeh- 
ner O.F.M. and E. Moody in the forefront. Today there is quite a 
group at work, though as yet a small one. 

But we are still at the beginning. Arabian and Jewish logic has 
hardly been touched ; texts and treatises are alike lacking. In the 
western domain some texts of Abelard have been published for the 
12th century; for the 14th and 15th centuries we have hardly 
anything, either new editions of texts or works on them; the 13th 
century is almost completely inaccessible and unknown. For this 
last, besides the (fairly) reliable older editions of the works of 
Thomas Aquinas and (some works) of Duns Scotus, Peter of Spain 
and William of Shyreswood are available only in provisional editions. 
For the 14th century there is an edition of the first book of Ockham's 
Summa, and one of a small work ascribed to Burleigh.* 

Altogether we must say that the present state of research permits 
no general survey of the sources, growth and details of scholastic 
logic. 

B. PROVISIONAL PERIODS 

However, on the basis of the works of Ph. Boehner, E. Moody, 
L. Minio-Paluello, and of the ever-growing number of general 
studies of medieval philosophy, the history of medieval logic can be 
provisionally divided into the following periods : 



* The late Fr. Ph. Boehner was working on a critical edition of another work 
of Burleigh's and of the Perutilis Logica of Albert of Saxony. 

148 



PERIODS 

1. transitional period: up to Abelard. So far as we know this is 
not remarkable for any logical novelties, and acquaintance even with 
earlier achievements was very limited. 

2. creative period: beginning seemingly after Abelard, about 
1150, and lasting to the end of the 13th century. Former achievement- 
now became known in the West, partly through the Arabs, partly (as 
L. Minio-Paluello has shown*) directly from Byzantium. At the 
same time work began on new problems, such as the proprieties 
lerminorum, properties of terms. By about 1260 the essentials of 
scholastic logic seem to have taken shape and been made widely 
known in text-books. The best known book of this kind, and the 
most authoritative for the whole of Scholasticism - though by no 
means the first or the only one - is the Summulae Logicales of 
Peter of Spain. 

3. period of elaboration: beginning approximately with William 
of Ockham (ob. 1349/50)** and lasting till the close of the Middle 
Ages. No essentially new problems were posed, but the old were 
discussed very thoroughly and very subtly, which resulted in an 
extremely comprehensive logic and semiotic. 

So little is known of the whole development that we are unable 

even to name only the most important logicians. We can only say 

with certainty that the following among others exercised great 

influence : 

in the 12th century: Peter Abelard (1079-1142); 

in the 13th century : Albert the Great (1 193-1280) ; 
Robert Kilwardby (ob. 1279), 
William of Shyreswood (ob. 1249), 
Peter of Spain (ob 1277); 

in the 14th century: William of Ockham (ob. 1349/50), 

John Buridan*** (ob. soon after 1358), 
Walter Burleigh (ob. after 1343), 
Albert of Saxony (1316-1390), 
Ralph Strode (ca. 1370) ; 

in the 15th century: Paul of Venice (ob. 1429), 

Peter Tarteret (wrote between 1480 and 

1490), 
Stephanus de Monte 

Appearance in this list comports no judgment of worth, especially 

as we hardly ever know whether a logician was original or only 

a copyist. 

* Verbal communication from Prof. L. Minio-Paluello to whom the author is 

obliged for much information about the 12th and beginning of the 13th century. 

** Ockham's productive period in logic was wholly prior to 1328/29 ^25.03). 

* * * Ph. Boehner states that John of Cornubia (Pseudo-Scotus) may belong to 

the same period. 

149 



SCHOLASTIC LOGIC 
C. THE PROBLEM OF SOURCES 

Even the question of the literary sources for Scholasticism's new 
logical problems is not yet satisfactorily answered. The works of 
Aristotle provide some starting-points for the semiotic, especially 
the first five chapters of the Hermeneia and the Sophistic Refutations. 
Recent research shows that the latter had a decisive influence on 
the scholastic range of problems*. But even the early scholastic 
theory of the 'properties of terms' is so much richer and more many- 
sided than the Aristotelian semiotic, that other influences must be 
supposed. Grammar was certainly an important one: so far as we 
can tell, that was the basis on which the main semiotic problems 
were developed without much outside influence - e.g. the whole 
doctrine of supposition, the growth of which can be traced with some 
continuity. 

We have no more certain knowledge about the origin of the 
'consequences'. Boethius's teaching about hypothetical sentences 
(rather than about hypothetical syllogisms) was undoubtedly very 
influential.** I.Thomas's recent inquiries (cf. footnote on 30.04) 
point to the Topics as a principal source; the Stoic fragments do not 
seem to have been operative, at least directly, although the Outlines 
of Pyrrhonism of Sextus Empiricus were already translated into 
Latin in the 14th century (25.04). We do find doctrines here and 
there which are recognizably Stoic, but in scholastic logical literature 
as a whole Stoic logic seems to have been known only in the (obscure) 
form of Boethius's syllogisms. But these do not underlie the con- 
sequences, since even in fairly late works the two are treated in 
distinction. Probably scholastic propositional logic is a rediscovery, 
starting from hints in the Topics and perhaps also the Hermeneia, 
rather than a continuation of Stoic logic. 

Arabian logicians certainly exercised some influence, though 
perhaps less than has commonly been supposed. But hardly any 
research has been done on this subject.*** 

D. LOGIC AND THE SCHOOLS 

The opinion has often been expressed in writings on the history 
of scholastic logic that it can be divided firstly according to schools, 
as it might be into nominalist and realist logic, secondly according 
to faculties, and so into an 'artistic' and a 'theological' logic. But 
these divisions are little relevant to formal logic as such. More 

* Verbal communication from Prof. Minio-Paluello. 
* * Prof. E. Moody has remarked on this to me. 
*** I. Madkour's VOrganon d'Aristote dans le monde Arabe (25.05) is quite 
inadequate. Prof. A. Badawi in Cairo has published and discussed a series of 
Arabic logical texts but unfortunately only in Arabic. Communications received 
from him indicate the presence of many interesting doctrines. 

150 



METHOD 

modern research has shown that a number of logicians belonging 
to sharply opposed philosophical schools, treated of just the same 
range of problems and gave the same answers. Thus in every case 
we have met there is but one doctrine of supposition, and differences 

are either to be ascribed to personal idiosyncrasy rather than 
philosophical presupposition, or else are more epistemological than 
logical. Any contrast between artistic and theological logic is hardly 
more in place. In the middle ages logic was always part of the 
curriculum of the faculty of arts, but no-one was admitted to the 
study of theology without having become Baccalaureun artium. 
Hence the chief theological works of this period presuppose and use 
the full range of 'artistic' logic. We should maintain only two 
distinctions relevant to this double division of logic: (1) the theolo- 
gians were not primarily interested in logic; (2) some of them 
elaborated logical doctrines of special importance for theology; an 
example is the doctrine of analogy of Thomas Aquinas. 

Thus in the Middle Ages we find essentially only one logic. 
Exceptions only occur where epistemological or ontological problems 
exert an influence, as in the determination of the notion of logic 
itself, and in the assigning of denotations. Everywhere else we find a 
unified logic, developing organically. The very multiplicity of medie- 
val views about extra-logical matters supports the thesis that 
formal logic is independent of any special philosophical position on 
the part of individual logicians. 

E. METHOD 

Our insufficient knowledge of the period makes it impossible to 
write a history of the evolution of its logic. A historical presentation 
would be possible for a few problems only, and even for those only 
for isolated spaces of time. The justification of this chapter in a 
work on the history of logical problems lies in the fact that, while 
un-historical in itself, it does to some extent exhibit one stage in 
the general development of logic. 

Two questions are raised by the choice of problems for discussion. 
The present state of research makes it likely that we are not ac- 
quainted with them all. In order not to miss at least the essentials, 
we have made great use of the Logica Magna of Paul of Venice, which 
expressly refers to all contemporary discussions and may rank as a 
veritable Summa of 14th century logic. Paul's range of problems 
has been enlarged by some further questions from other authors. 

The second difficulty is posed by those logical problems which 
overlap epistemology and methodology. Aristotle and the thinkers 
of the Megarian-Stoic school envisage them in a fairly simple way, 
but scholastic conceptions and solutions are much more complicated. 
In order not to overstep our limits too far, these matters will be 
touched on only very superficially. 

151 



SCHOLASTIC LOGIC 
F. CHARACTERISTICS 

A survey of the logical problems dealt with by the Scholastics 
clearly shows that they fall into two classes : on the one hand there 
are the ancient ones, Aristotelian or Megarian-Stoic, concerning 
e.g. categorical and modal syllogistic, hypothetical syllogisms (i.e. 
Stoic arguments) etc. The rest, on the other hand, are either quite 
new, or else presented in so new a guise as no longer to remind one 
of the Greeks. Conspicuous in this class are the doctrines of 'proper- 
ties of terms', of supposition, copulation, appellation and amplia- 
tion, then too the doctrine of consequences which while dependent 
on Aristotle's Topics and the Stoics, generalizes the older teaching 
and puts it in a new perspective. The same must be said about the 
insolubles (§ 35) which treat of the Liar and such-like but by new 
methods and in a much more general way. 

Generally speaking, whatever the Scholastics discuss, even the 
problems of antiquity, is approached from a new direction and by 
new means. This is more and more the case as the Middle Ages 
progress. There is firstly the metalogical method of treatment. 
Metalogical items are indeed to be found in Aristotle (14.85 fif.), 
but in Scholasticism, at least in the later period, there is nothing 
but metalogic, i.e. formulae are not exhibited but described, so 
that in many works, e.g. in the De purilaie artis logicae of Burleigh 
not a single variable of the object language is to be found. Even 
purely Aristotelian matters such as the categorical syllogism are 
dealt with from the new points of view, semiotic and other. In early 
Scholasticism a double line of development is detectable, problems 
inherited from antiquity being treated in the spirit of the ancient 
logicians, as in the commentaries of Albert the Great, and the new 
doctrine being developed in the very same work. Later the latter 
becomes more and more prominent, so that, as has been said, even 
genuinely Aristotelian problems are presented metalogically, in 
terms of the doctrine of supposition etc. 

In addition, scholastic logic, even by the end of the 13th century, 
is very rich, very formalistic and exact in its statement. Some 
treatises undoubtedly rank higher than the Organon and perhaps 
than the Megarian-Stoic fragments too. The title of Burleigh's 
work - 'De purilaie arlis logicae' - suits the content, for here is a 
genuinely pure formal logic. 



152 



I. SEMIOTIG FOUNDATIONS 

§26. SUBJECT-MATTER OF LOGIC 

To be able to understand what the Scholastics thought logic was 
about, one must be acquainted with the elements of their semiotic. 
Hence we give first two texts from Peter of Spain followed by one 
from Ockham, about sounds and terms. 

A. BASIC NOTIONS OF SEMIOTICS 

26.01 A sound is whatever is properly perceived by hearing; 
for though a man or a bell may be heard, this is only by means 
of sound. Of sounds, one is voice, another not voice. Sound- 
voice is the same as voice; so voice is sound produced from 
the mouth of an animal, formed by the natural organs. . . . 
Of voices, one is literate, another not literate. Literate voice 
is that which can be written, e.g. 'man'; not literate is that 
which cannot be written. Of literate voices one is significant, 
another not significant. Significant voice is that which 
represents something to the hearing, e.g. 'man' or the groans 
of the sick which signify pain. Not significant voice is that 
which represents nothing to the hearing, e.g. 'bu', 'ba'. Of 
significant voices one signifies naturally, another convention- 
ally. Conventionally significant voice is that which represents 
something at the will of one who originates it, e.g. 'man'. 
Naturally significant voice is that which represents the same 
thing to all, e.g. the groans of the sick, the bark of dogs. Of 
conventionally significant voices one is simple or not complex, 
e.g. a noun or a verb, another composite or complex, e.g. a 
speech (oratio). . . . 

And it should be known that logicians (dialedicus) posit 
only two parts of speech, viz. noun and verb , calling the others 
'syncategoremata'. 

26.02 Of things which are said, some are said with com- 
plexity, e.g. 'a man runs', 'white man'. Others without 
complexity, e.g. 'man' by itself, a term that is not com- 
plex. ... A term, as here understood, is a voice signifying a 
universal or particular, e.g. 'man' or 'Socrates'. 

These texts contain doctrine generally accepted in Scholasticism. 
Another, no less widely recognized, is excellently summarized by 
Ockham, who uses the expression 'terminus conceptus' ('conceived 
term') instead of the usual 'terminus mentalis' ('thought term'). 

153 



SCHOLASTIC LOGIC 

26.03 It is to be known that according to Boethius . . . 
speech is threefold, viz. written, spoken and conceived, this 
last having being only in the intellect, so (too) the term is 
threefold, viz. written, spoken and conceived. A written term 
is part of a proposition written down on some body which 
is seen or can be seen by a bodily eye. A spoken term is part 
of a proposition spoken by the mouth and apt to be heard 
with a corporeal ear. A conceived term is an intention or 
affection of the soul, naturally signifying something or con- 
signifying, apt to be part of a proposition in thought. . . . 

Those are the most important presuppositions for what follows. 

B. LOGIC AS A THEORY OF SECOND INTENTIONS 

Many early Scholastics give explicit definitions of logic. Disregard- 
ing these, we shall proceed to descriptions of the subject-matter of 
logic, of which we know two kinds. According to the first it consists 
in so-called second intentions. Three series of texts will illustrate the 
matter, taken from Thomas Aquinas (13th century), Ockham and 
Albert of Saxiony (early and late 14th century respectively). 

26.04 Being is two-fold, being in thought (ens rationis) and 
being in nature. Being in thought is properly said of those 
intentions which reason produces (adinvenit) in things it 
considers, e.g. the intention of genus, species and the like, 
which are not found among natural objects, but are consequent 
on reason's consideration. This kind, viz. being in thought, 
is the proper subject-matter of logic. 

26.05 The relation which is denoted (importatur) by this 
name 'the same' is merely a being in thought, if what is the 
same without qualification is meant: for such a relation can 
only consist in an ordering by the reason of something to 
itself, according to some two considerations of it. 

26.06 Because relation has the weakest being of all the 
categories, some have thought that it belongs to second 
intentions (intelledibus). For the first things understood are 
the things outside the soul, to which the intellect is primarily 
directed, to understand them. But those intentions (inten- 
tiones) which are consequent on the manner of understanding 
are said to be secondarily understood. ... So according to 
this thesis (positio) it would follow that relation is not among 
the things outside the soul but merely in the intellect, like the 

154 



SEMIOTICS 

intention of genus and species and second (i.e. universal) 
substances. 

Thus according to Thomas the subject-matter of logic is such 
'secondarily understood things' or 'second intentions', belonging 
to the domain of being in thought, and so lecta. Not all lecla, however, 
but a special kind, such as those corresponding to the meaning of 
logical constants. It is to be stressed that according to Thomas, 
as for the Stoics, the subject-matter of logic is nothing psychical, 
but something objective, which yet exists only in the soul. 

The nature of second intentions war much debated among 
Scholastics, and we know of many different opinions. Ockham says: 

26.07 It should first be known that that is called an 'inten- 
tion of the soul' which is something in the soul apt to signify 
something else. . . . But what is it in the soul which is such 
a sign? It must be said that on that point (articulum) there 
are various opinions. Some say that it is only something 
fashioned by the soul. Others that it is a quality subjectively 
existing in the soul, distinct from the act of understanding. 
Others say that it is the act of understanding. . . . These 
opinions will be examined later. For the present it is enough 
to say that an intention is something in the soul which is a 
sign naturally signifying something for which it can stand 
(supponere) or which can be part of a mental proposition. 

Such a sign is twofold. One which is a sign of something 
which is not such a sign, . . . and that is called a 'first intention' 
such as is that intention of the soul which is predicable of 
all men, and similarly the intention predicable of all white- 
nesses, and blacknesses, and so on. . . . But a second intention 
is that which is a sign of such first intentions, such as are the 
intentions 'genus', 'species' and such-like. For as one intention 
common to all men is predicated of all men when one says: 
'this man is a man', 'that man is a man', and so on of each 
one, similarly one intention common to those intentions 
which signify and stand for things is predicated of them 
when one says: . . . 'stone is a species', 'animal is a species, 
'colour is a species' etc. 

The same doctrine is further developed by Albert of Saxony: 

26.08 'Term of first intention' is the name given to that 
mental term which is significative of things not from the 
point of view of their being signs. Thus this mental term 
'man', or this mental term 'being', or this mental term 'qua- 

155 



SCHOLASTIC LOGIC 

lity', or this mental term 'voice'. Hence this mental term 
'man' signifies Socrates or Plato, and not insofar as Socrates 
or Plato are signs for other things. . . . But a mental term 
which is naturally significative of things insofar as they are 
signs is called a 'term of second intention', and if they ceased 
to be signs it would not signify them. Of this kind are the 
mental terms 'genus', 'species', 'noun', 'verb', 'case of a noun' 
etc. 

In the last two texts the conception is other than that of Thomas. 
Second intentions are there conceived in a purely semantic way; 
they are signs of signs, and for Albert signs of signs as such. 

Whether Ockham and Albert thought of logic as in any sense a 
science of second intentions remains open to question. One might 
perhaps give expression to both their views by saying that logic 
is a science constructed throughout in a meta-language,* remarking 
at the same time that the Scholastics included under 'signs' mental 
as well as exterior (written or spoken) signs. 

However, one common feature underlies all these fundamental 
differences; logic is sharply distinguished from ontology in the 
whole scholastic tradition. This is so for Thomas, since its object 
is not real things, but second intentions; and for his successors, since 
it is expressed not in an object- but in a meta-language. 

It should also be noted that in fact the entire practice of medieval 
logic corresponds to the Thomist conception of the object of logic, 
even though this conception was not the only one. For scholastic 
logic essentially consists of two parts: the doctrine of the properties 
of terms, and the doctrine of consequences. The properties of terms 
are evidently second intentions in the Thomist sense; and one must 
think oi consequences in the same light, since the logical relationships 
they exhibit (e.g. between antecedent and consequent) are not real 
things. 

C. FORMAL LOGIC AS A THEORY OF 
SYNCATEGOREMATIC EXPRESSIONS 

There is a difficulty in adopting the view that we have hypotheti- 
cally ascribed to Ockham about the subject matter of logic, in that 
it does not achieve a definition of logic as a distinct science, since 
every science can be formulated in a meta-language. But we find, 
though not explicitly, logic limited to concern with logical form, 
which leads to an exact definition of formal logic when this form 
is equated with the syncategoremata. Scholastic practice is wholly in 
accord with this definition in its cultivation of the corresponding 

* I am particularly obliged to Prof. E. Moody for valuable assistance with 
these questions. 

156 



SEMIOTICS 

theory of logical form. Three texts about Byncategoremata follow, 
one from William of Shyreswood (13th century), one from Ockham 
(beginning of 14th) and one from Buridan* (end of 14th). 

26.09 To understand propositions one must know their 
parts. Their parts are twofold, primary and secondary. 
Primary parts are substantival names and verbs; these are 
necessary for an understanding of propositions. Secondary 
parts are adjectival names, adverbs, conjunctions and prepo- 
sitions; these are not essential to the constitution of proposi- 
tions. 

Some secondary parts are determinations of primary ones 
with reference to (ratione) their things (i.e. to which they 
refer), and such are not syncategoremata ; e.g. when I say 
'white man' 'white' signifies that one of its things, a man, is 
white. Others are determinations of primary parts insofar as 
these are subjects or predicates; e.g. when I say 'every man 
runs', the 'every', which is a universal sign, does not mean 
that one of its things, namely a man, is universal, but that 
'man' is a universal subject. Such are called 'syncategoremata' 
and will be treated (here), as offering considerable difficulties 
in discourse. 

Ockham affords a development of the same thought: 

26.10 Categorematic terms have a definite and certain 
signification, e.g. this name 'man' signifies all men, and this 
name 'animal' all animals, and this name 'whiteness' all 
whitenesses. But syncategorematic terms, such as are 'all', 
'no', 'some', 'whole', 'besides', 'only', 'insofar as' and such- 
like, do not have a definite and certain signification, nor do 
they signify anything distinct from what is signified by the 
categoremata. Rather, just as in arabic numeration a zero 
(cifra) by itself signifies nothing, but attached to another 
figure makes that signify, so a syncategorema properly speak- 
ing signifies nothing, but when attached to something else 
makes that signify something or stand for some one or more 
things in a determinate way, or exercises some other function 
about a categorema. Hence this syncategorema 'all' has no 
definite significate, but when attached to 'man' makes it 

* It is taken from the Consequentiae which is ascribed to Buridan in the early 
printed editions, though a letter from Fr. Ph. Boehner informs us that no MS of 
this work has yet been found. 

157 



SCHOLASTIC LOGIC 

stand or suppose for all men . . . , and attached to 'stone' 
makes it stand for all stones, and attached to 'whiteness' 
makes it stand for all whitenesses. And the same is to be held 
proportionately for the others, as for that syncategorema 
'all', though distinct functions are exercised by distinct 
syncategoremata, as will later be shown for some of them. 

Evidently, the syncategoremata are our logical constants. That 
they determine logical form is expressly and consciously propound- 
ed by Buridan (whose text was later adopted almost word for 
word by Albert of Saxony: 26.11). 

26.12 When form and matter are here spoken of, by the 
matter of a proposition or consequence is understood merely 
the categorematic terms, i.e. the subject and predicate, to 
the exclusion of the syncategorematic * ones attached to 
them, by which they are restricted, negated, or divided and 
given (trahuntur) a determinate kind of supposition. All else, 
we say, belongs to the form. Hence we say that the copula, 
both of the categorical and of the hypothetical proposition 
belongs to the form of the proposition, as also negations, 
signs, the number both of propositions and terms, as well 
as the mutual ordering of all the aforesaid, and the inter- 
connections of relative terms and the ways of signifying 
(modos significandi) which relate to the quantity of the 
proposition, such as discreteness**, universality etc. . . . 

E.g. . . Since modals have subordinate copulas and so differ 
from assertoric propositions, these differ in form ; and by reason 
of the negations and signs (signa) affirmatives are of another 
form than negatives, and universals than particulars; and by 
reason of the universality and discreteness*** of their terms 
singular propositions are of another form than indefinites ; by 
reason of the number of terms the following propositions are 
of different forms: 'man is man' and 'man is ass', as are the 
following consequences or hypothetical propositions: 'every 
man runs, therefore some man runs' and 'every man runs, 
therefore some ass walks about'. Similarly by reason of the 
order the following are of different forms: 'every man is 
animal', 'animal is every man', and likewise the following 



* Reading syncategoremaiicis for categoricis. 
* * Reading discretio for descriptio. 
* * * See last note. 

158 



CONTENT 

consequences: 'every B is A, therefore some B is A 1 and 
'every B is A, therefore some A is B' etc. Similarly by rea- 
son of the relationship and connection . . . 'the man runs, the 
man does not run' is of another form than this: 'the man runs 
and the same does not run': since its form makes the second 
impossible, but it is not so with the first. 

It is easy to establish that scholastic logic has for its object 
precisely form so conceived. The doctrine of the properties of terms 
treats of supposition, appellation, ampliation and such-like rela- 
tionships, all of which are determined in the proposition by syn- 
categorematic terms; while the second part of scholastic logic, 
comprising the doctrine of the syllogism, consequences etc., treats 
of formal consequence, which holds in virtue of the form as de- 
scribed. 

Expressed in modern terms, the difference between the two 
conceptions of logic that have been exemplified, is that the first is 
semantic, the second syntactical: for the first uses the idea of 
reference, the second determines logical form in a purely structural 
way. According to the second the logical constants are the subject- 
matter of logic, while on the view of Thomas this object is their 
sense. On either view Scholasticism achieved a very clear idea of 
logical form and so of logic itself. 

D. CONTENT OF THE WORKS 

Two kinds of logical works can be distinguished in Scholasticism, 
commentaries on Aristotle and independent treatises or manuals. 
To begin with, the composition of works even of the second kind 
is strongly influenced by the Aristotelian range of problems, at least 
in the sense that newer problems are incorporated into the frame- 
work of the Organon. It is only gradually that the ever growing 
importance of the new problems finds expression in the very con- 
struction of the works. We shall show this in some examples collected 
for the most part by Ph. Boehner (26.13). 

Albert the Great has no independent arrangement; his logic 
consists of commentaries on the writings of Aristotle and Boethius. 

The chief logical work of Peter of Spain falls into two parts; the 
first is markedly Aristotelian and contains the following treatises: 
On Propositions (= Hermeneia), 
On the Predicables (= Porphyry), 
On the Categories (= Categories), 
On Syllogisms (= Prior Analytics), 
On Loci (= Topics), 
Suppositions, 
On Fallacies (= Sophistic Refutations). 

159 



SCHOLASTIC LOGIC 

In the second part is to be found nothing but the new problematic, 
for it is divided into treatises on 

Relatives, 

Ampliations, 

Appellations, 

Restrictions, 

Distributions. 

Two points are notable: that propositions are discussed at the start 
(and not in the third place as in Porphyry and the Categories), and 
that the doctrine of supposition is inserted before the treatise on 
fallacies. That shows how the new problematic began to influence 
the older one. 

Ockham's Summa is divided in another way: 
I. Terms: 

1. In general. 

2. Predicables. 

3. Categories. 

4. Supposition. 

II. Propositions: 

1. Categorical and modal propositions. 

2. Conversion. 

3. Hypothetical propositions. 

III. Arguments: 

1. Syllogisms: 

a) assertoric, 

b) modal, 

c) mixed (from the first two kinds), 

d) 'exponibilia', 

e) hypothetical. 

2. Demonstration (in the sense of the Posterior Analytics). 

3. Further rules: 

a) Consequences. 

b) Topics. 

c) Obligations. 

d) Insolubles. 

4. Sophistics. 

The general framework here is still Aristotelian, more so even than 
with Peter, but the new problematic has penetrated into the 
subdivisions. An Aristotelian title often conceals strange material, 
as when the chapter on the categories deals with typically scholastic 
problems about intentions etc. 

160 



CONTENT 

Walter Burleigh's De purilale artis logicae is divided thus: 
I. On terms: 

1. Supposition. 

2. Appellation. 

3. Copulation. 

II. (Without title): 

1. Hypothetical propositions. 

2. Conditional syllogisms. 

3. Other hypothetical syllogisms. 

Even this small sample shows how the scholastic range of problems 
is to the fore. 

Albert of Saxony divides his logic in this way: 

1. Terms (in general). 

2. Properties of terms (supposition, ampliation, appellation). 

3. Propositions. 

4. Consequences: 

a) in general. 

b) Propositional consequences. 

c) Syllogistic consequences. 

d) Hypothetical syllogisms. 

e) Modal syllogisms. 

f) Topics. 

5. Sophistics. 

6. Antinomies and obligations. 

Here the whole of Aristotelian and Stoic formal logic has been built 
into the scholastic doctrine of consequences, while this last is 
introduced by discussion of another typically scholastic matter, 
the properties of terms. 

Finally we consider the division of the Logica Magna of Paul 
Nicollet of Venice (ob. 1429), which is probably the greatest syste- 
matic work on formal logic produced in the Middle Ages. It falls 
into two parts, the first designed to treat of terms, the second of 
propositions, though in fact the first contains much about propo- 
sitions, and the second includes also the doctrine of consequences 
and syllogisms. 

Part I : 

1. Terms. 

2. Supposition. 

3. Particles that cause difficulty. 

4. Exclusive particles. 

5. Rules of exclusive propositions. 

6. Exceptive particles. 

7. Rules of exceptive propositions. 

8. Adversative particles. 

161 



SCHOLASTIC LOGIC 

9. 'How'. 

10. Comparatives. 

11. Superlatives. 

12. Objections and counter-arguments. 

13. Categorematic 'whole' (totus). 

14. 'Always' and 'ever'. 

15. 'Infinite'. 

16. 'Immediate'. 

17. 'Begins' and 'ceases'. 

18. Exponible propositions. 

19. Propositio officiabilis. 

20. Composite and divided sense. 

21. Knowing and doubting. 

22. Necessity and contingence of future events. 

Part II : 

1. Propositions (in general). 
2.-3. Categorical propositions. 

4. Quantity of propositions. 

5. Logical square. 

6. Equivalences. 

7. Nature of the proposition in the square. 

8. Conversion. 

9. Hypothetical propositions. 

10. Truth and falsity of propositions. 

11. Signification of propositions. 

12. Possibility, impossibility. 

13. Syllogisms. 

14. Obligations. 

15. Insolubles. 

Here the treatise on consequences has disappeared, having been 
incorporated into that on hypothetical propositions.* 



§27. SUPPOSITION 

We begin our presentation of scholastic logic with the doctrine 
of supposition. This is one of the most original creations of Schola- 
sticism, unknown to ancient and modern logic, but playing a 

* The following figures will give an idea of the scope of this work. The Logica 
Magna occupies 199 folios of four columns each containing some 4600 printed 
signs, so that the whole work comprises about 3,650,000 signs. This corresponds 
to at least 1660 normal octavo pages, four to five volumes. But the Logica Magna 
is only one of four works by Paul on formal logic, the others together being even 
more voluminous. None of it is merely literary work, but a pure logic, written in 
terse and economical language. 

162 



SUPPOSITION 

central role here. Unpublished research of L. Minio Paluello enables 

us to trace its origin to the second hall of the 12th century. By the 
middle of the 13th all available sources witness to its being every- 
where accepted. Later there appear some developments of detail, 
but no essentially new fundamental ideas. 

We shall first illustrate the notion of supposition in general, then 
proceed to the theory of material and simple supposition, and finally 
mention other kinds. 



A. CONCEPT OF SUPPOSITION 

The notion of supposition is already well defined in Shyreswood, 
and distinguished by him from similar 'properties of terms' : 

27.01 Terms have four properties, which we shall now 
distinguish. . . . These properties are signification, supposition, 
copulation and appellation. Signification is the presentation 
of a form to the reason. Supposition is the ordering of one 
concept (intelledus) under another. Copulation is the ordering 
of one concept over another. It is to be noted that supposition 
and copulation, like many words of this kind, are proffered 
(dicuntur) in two senses, according as they are supposed to be 
actual or habitual. Their definitions belong to them according 
as they are supposed to be actual. But insofar as they are 
supposed to be habitual, 'supposition' is the name given 
to the signification of something as subsisting; for what 
subsists is naturally apt to be ordered under another. And 
'copulation' is the name given to the signification of some- 
thing as adjacent, for what is adjacent is naturally apt to be 
ordered over another. But appellation is the present attribution 
of a term, i.e. the property by which what a term signifies can 
be predicated of something by means of the verb 'is'. 

It follows that signification is present in every part of 
speech, supposition only in substantives, pronouns or sub- 
stantival particles; for these (alone) signify the thing as 
subsistent and of such a kind as to be able to be set in order 
under another. Copulation is in all adjectives, participles and 
verbs, appellation in all substantives, adjectives and parti- 
ciples, but not in pronouns since these signify substance only, 
not form. Nor is it in verbs. . . . None of these three, supposi- 
tion, copulation and appellation is present in the indeclinable 
parts (of speech), since no indeclinable part signifies substance 
or anything in substance. 

163 



SCHOLASTIC LOGIC 
Thomas Aquinas speaks in similar fashion: 

27.02 The proper sense (ratio) of a name is the one which 
the name signifies; . . . But that to which the name is attri- 
buted if it be taken directly under the thing signified by the 
name, as determinate under indeterminate, is said to be 
supposed by the name; but if it be not directly taken under 
the thing of the name, it is said to be copulated by the name; 
as this name 'animal' signifies sensible animate substance, 
and 'white' signifies colour disruptive of sight, while 'man' 
is taken directly under the sense of 'animal' as determinate 
under indeterminate. For man is sensible animate substance 
with a particular kind of soul, viz. a rational one. But it is 
not directly taken under white, which is extrinsic to its essence. 

27.03 The difference between substantives and adjectives 
consists in this, that substantives refer to (ferunt) their 
suppositum, adjectives do not, but posit in the substance* 
that which they signify. Hence the logicians (sophistae) say 
that substantives suppose, adjectives do not suppose but 
copulate. 

The doctrine implicit in these texts was later expressly formulated 
by Ockham: 

27.04 (Supposition) is a property belonging to terms, but 
only as (they occur) in a proposition. 



B. MATERIAL AND FORMAL SUPPOSITION 

Shyreswood writes: 

27.05 Supposition is sometimes material, sometimes formal. 
It is called material when an expression (diciio) stands either 
for an utterance (vox) by itself, or for the expression which 
is composed of an utterance and (its) significance, e.g. if 
we were to say: 'homo' consists of two syllables, 'homo' is 
a name. It is formal when an expression stands for what it 
signifies. 

27.06 The first division of supposition is disputed. For it 
seems that kinds not of supposition but of signification are 
there distinguished. For signification is the presentation of a 
form to the reason. So that where there is different presenta- 

Reading substantiam for substantivum. 

164 



SUPPOSITION 

tion there is different signification. Now when an expression 
supposes materially it presents either itself or its utterance; 
but when formally, it presents what it signifies; therefore it 
presents something different (in each case); therefore it 
signifies something different. But that is not true, since 
expressions by themselves always present what they signify, 
and if they present their utterance they do not do this of 
themselves but through being combined with a predicate. For 
some predicates naturally refer to the mere utterance or to 
the expression, while others refer to what is signified. But this 
effects no difference in the signification. For the expression as 
such, before ever being incorporated in a sentence, already 
has a significance which does not arise from its being co- 
ordinated with another. 

On this question Thomas Aquinas remarks: 

27.07 One could object to this (teaching of ours) also, 
that verbs in other moods (than the infinitive) seem to be put 
as subjects, e.g. if one says: 'I run is a verb'. But it must be 
said that the verb 'I run' is not taken formally in this state- 
ment (locutio). (i.e.) with its signification referred to a thing, 
but as materially signifying the word itself which is taken as a 
thing. 

The expressions 'suppositio materialis' and 'suppositio formalis' 
have also another meaning for Thomas. He sometimes uses the 
first for suppositio personalis (cf. 27.23 ff.) and the second for 
suppositio simplex (cf. 27.17f.) : 

27.08 A term put as subject holds (tenetur) materially, 
i.e. (stands) for the suppositum; but put as predicate it 
holds formally, i.e. (stands) for the nature signified. 

Perhaps this ambiguity accounts for the expression 'formal 
supposition', that we have found in Shyreswood and Thomas, later 
disappearing, so far as we know, outside the Thomist school.* 
Even by Ockham's time supposition is divided immediately into 
three kinds: 

27.09 Supposition is first divided into personal, simple 
and material. 



*Fr. Ph. Boehner is to be thanked for the information that this expression 
occurs in Chr. Javellus (ob. 1538). 

165 



SCHOLASTIC LOGIC 

The two first of those are sub-species of the formal supposition of 
Shyreswood and Thomas, which Ockham no longer refers to. His 
division is subsequently the usual one, except among the Thomists. 
In 27.05 we read of an 'utterance by itself and an 'expression 
which is composed of an utterance and (its) significance'. This 
distinction is developed at the end of the 15th century by Peter 
Tarteret: 

27.10 Material supposition is the acceptance of a term for 

its non-ultimate significate, or its non- ultimate significates 

In which it is to be noticed that significates are two-fold, 
ultimate and non-ultimate. The ultimate significate is that 
which is ultimately signified by a term signifying conven- 
tionally, and ultimately or naturally and properly. But the 
non-ultimate significate is the term itself, or one vocally or 
graphically similar, or one mentally equivalent. From which 
it follows that a vocal or written term is said to signify 
conventionally in two ways, either ultimately or non-ulti- 
mately. Ultimately it signifies what it is set to signify; but 
a vocal term is said to signify conventionally and non-ulti- 
mately a synonymous written term; and a written term 
is said to signify non-ultimately an utterance synonymous 
with it. . . . 

From the modern point of view this doctrine reflects our distinc- 
tion of language and meta-language, except that in place of two 
languages, symbols of one language exercise a two-fold supposition. 
Furthermore, the two last-cited texts exhibit the important distinc- 
tion between the name of an individual symbol and the name of a 
class of equiform symbols. We do not find this in the logistic period 
till after 1940. 

This distinction first occurs, so far as we know, in St. Vincent 
Ferrer* (14th century), as a division of material supposition: 

27.11 Material supposition is divided as is formal. One 
(kind of material supposition) is common, the other discrete. 

* Vincent Ferrer was the greatest preacher of his time. We would add that 
Savonarola was also an important logician. A similar link between deep religiDUS 
life and a talented interest in formal logic is also to be observed in Indian culture 
especially among the Buddhists. This would seem to be a little known and as yet 
unexplained phenomenon. The authenticity of Vincent Ferrer's philosophical 
opuscules De Suppositionibus and De imitate universalis has been challenged so 
far as we know only by S. Brettle (vid. Additions to Bibliography 3.98). To his 
p. 105 note 3 should be added a, here relevant, reference to p. 33 note 10. M. G. 
Miralles (vid. Additions) summarizes the arguments for and against, and 
concludes with M. Gorce (vid. Additions): 'L'authenticite des deux 6crits n'a ete 
jamais mise en doute. Le temoignage du contemporain Ranzzano suffit a la 
prouver.' 

166 



SUPPOSITION 

It is discrete if the term or utterance stands determinately 
for a suppositum of its material significate. And thus discrete 
material* suppositum occurs in three ways. In one way 
through the utterance or term itself, as when one asks: 
'What is it you want to say?' and the other answers: 'I say 
"buf" and '"baf" is said by me', (then) the subject of this 
proposition supposes materially and discretely since it stands 
for the very utterance numerically identical (with it) (cf. 
11.11). This becomes more evident if names are assigned to 
the individual terms in such a way that as this name 'man' * 
signifies this individual man so this name l A' signifies that 
individual word 'buf' and '£' the other ('baf'). And then if 
it is said: l A is an utterance' or 'A is said by me', the 
subject supposes materially and discretely, as in the proposi- 
tion 'Socrates runs' the subject supposes formally and dis- 
cretely. 

27.12 It occurs secondly through a demonstrative name 
(nomen) demonstrating an utterance or singular term, as 
when the utterance of the term 'man' is written somewhere 
and one says, with reference to this utterance: 'That is a 
name'. Then the subject of the proposition supposes materially 
for that which it demonstrates. 

It occurs in a third way through a term. . . ., which is 
determined by a demonstrative pronoun, as when it is said 
of the written utterance 'man': 'this "man" is a name' or 
'this utterance is a name'. 

And each of these ways . . . can be varied by natural, 
personal or simple supposition, as was said about singular 
formal supposition. 

Common (communis) material supposition is when the 
utterance or term stands indeterminately for its material 
signification, as when it is said: "people" is written' the sub- 
ject of this proposition stands indeterminately for this term 
'people', or (in another example) for some other (term). I 
do not say that in the proposition '"people" is written' or in 
some other such that the supposition is indeterminate, but 
that the subject is indeterminate and is taken indeterminately 

Material common supposition is divided into natural, 
personal and simple supposition, like common formal supposi- 

* Reading materialis for formalis. 
** Reading homo for primo. 

167 



SCHOLASTIC LOGIC 

tion. . . . An example of personal : ' "man" is heard', ' "man" is 
written ',' "man" is answered'. An example of simple 
(supposition): '"man" is a species of utterance', '"man" is 
conceived', '"man" is said by this man'; and so on in many 
other cases as everyone can see for himself. 

So material supposition is divided just as is formal. These texts 
exhibit scholastic semantics at its best. This accuracy of analysis 
is the more astonishing when one remembers that the distinction 
mentioned in the introduction to 27.11 remained unknown, not 
only to the decadent 'classical' logicians, but also to mathematical 
ones for nearly a century. 

It should also be noticed that in the text of Tarteret just cited, a 
distinction occurs which cannot be expressed in contemporary 
terms. The Scholastics distinguished, as has already be said above 
(26.03) three inter-related kinds of sign: graphical, vocal and 
psychic, and a materially supposing graphical sign can stand either 
for itself (or its equiforms) or for the corresponding vocal or psychic 
sign. 

Burleigh has another division, parallel to that between material 
and formal supposition: 

27.13 The tenth rule is: that on every act that is accom- 
plished there follows the act that is signified, and conversely. 
For it follows: 'man is an animal, therefore "animal" is 
predicated of "man"', for the verb 'is' accomplishes predica- 
tion, and this verb 'is predicated' signifies predication, and 
syncategorematic particles accomplish acts, and adjectival 
verbs signify such acts. E.g. the sign 'all' accomplishes 
distribution, and the verb 'to distribute' signifies distributions; 
the particle 'if exercises consequence, and this verb 'it follows' 
signifies consequence. 

It was said above that this distinction runs parallel with that 
between formal and material supposition, for it could easily be 
translated into it. But Burleigh would not seem to be thinking of 
these suppositions here; by 'the act signified' he means not words, 
but their significates. For in his example, the word 'animal' is not 
predicated of the word 'man', but what the first signifies is predi- 
cated of that for which the second supposes. 

C. SIMPLE SUPPOSITION 

Along with the idea of material supposition, that of simple 
(simplex) supposition is an interesting scholastic novelty. On this 
subject we can limit ourselves to the 13th century, and mainly to 

168 



SUPPOSITION 

Peter of Spain. First we shall give some of his general divisions of 
formal supposition : 

27.14 One kind of supposition is common, another discrete. 
Common supposition is effected by a common term such as 
'man'. Again of common suppositions one kind is natural, 
another accidental. Natural supposition is the taking of a 
common term for everything of which it is naturally apt to be 
predicated, as 'man' taken by itself naturally possesses 
supposition for all men who are and who have been and who 
will be. Accidental supposition is the taking of a common 
term for everything for which its adjunct requires (it to be 
taken). E.g. 'A man exists'; the term 'man' here supposes for 
present men. But when it is said: 'a man was', it supposes for 
past men. And when it is said : 'a man will be', it supposes for 
future ones, and so has different suppositions according to the 
diversity of its adjuncts. 

Later on we also meet with an 'improper' (27.15) and a 'mixed' 
(27.16) supposition. The first simply consists in the metaphorical 
use of a term. The second was introduced to elucidate the function 
of terms of which one part supposed in one way, another in another. 
From the logical point of view these are not very important ideas. 
Of greater importance is Peter's continuation : 

27.17 Of accidental suppositions one is simple, another 
personal. Simple supposition is the taking of a common term 
for the universal thing symbolized (figurata) by it, as when it 
is said : 'man is a species' or 'animal is a genus', the term 'man' 
supposes for man in general and not for any of its inferiors, 
and similarly in the case of any common term, as 'risible is a 
proprium', 'rational is a difference'. 

27.18 Of simple suppositions one belongs to a common 
term set as subject, as 'man is a species'; another belongs to a 
common term set as an affirmative predicate, as 'every man 
is an animal'; the term 'animal' set as a predicate has simple 
supposition because it only supposes for the generic nature; 
yet another belongs to a common term put after an exceptive 
form of speech, as 'every animal apart from man is irrational'. 
The term 'man' has simple supposition. Hence it does not 
follow: 'every animal apart from man is irrational, therefore 
every animal apart from this man (is irrational)', for there 
is there the fallacy of the form of speech (cf. 11.19), when 
passage is made from simple to personal supposition. Similarly 

169 



SCHOLASTIC LOGIC 

here: 'man is a species, therefore some man (is a species)'. In 
all such cases passage is made from simple to personal sup- 
position. 

27.19 But that a common term put as predicate is to be 
taken with simple supposition is clear when it is said : 'of all 
contraries there is one and the same science', for unless the 
term 'science' had simple supposition there would be a fallacy. 
For no particular science is concerned with all contraries; 
medicine is not concerned with all contraries but only with 
what is healthy and what is sick, and grammar with what is 
congruous and incongruous, and so on. 

This is to be compared with the text of Thomas cited above 
(27.08). The following text from him from a theological context 
expresses the matter clearly: 

27.20 The proposition homo fadus est Deus . . . can be 
understood as though fadus determines the composition, so 
that the sense would be: 'a man is in fact God', i.e. it is a fact 
that a man is God. And in this sense both are true, homo fadus 
est Deus and Deus fadus est homo. But this is not the proper 
sense of these propositions (locutionum), unless they were to be 
so understood that 'man' would have not personal but simple 
supposition. For although this (concrete) man did not become 
God, since the suppositum of this, the person of the Son of 
God, was God from eternity, yet man, speaking universally, 
was not always God. 

This text has the further importance that it may suggest the rea- 
son why the Scholastics spoke of 'personal' supposition, this being 
the function exercised by a term in standing for individuals or an 
individual (suppositum). For this recalls to the mind of a Scholastic 
the famous theological problem of the person of Christ, as in 27.20. 

The essentials of the scholastic doctrine of simple supposition 
may be summed up thus: in the proposition l A is B', the subject 'A' 
has of itself personal supposition, i.e. it stands for the individuals, 
but the predicate l B' has simple supposition, i.e. it stands either for 
a property or a class. But one can also frame propositions in which 
something is predicated of such a property or class, and then the 
subject must have simple supposition. It can be seen that this 
doctrine deals with no less a subject than the distinction between 
two logical types, the first and second (cf. 48.21). 

These simple but historically important facts are complicated by 
the scholastic development of two other problems along with this 
doctrine. They are (1) the problem of analysing propositions, 

170 



SUPPOSITION 

whether they should be understood in a purely extensions! fashion, 
or with extensional subject and intensional predicate. Thomas and 
Peter, in the texts cited, adopt the second position. We shall treat 
this problem a little more explicitly in a chapter on the analysis 
of propositions (29.02-04). Then (2) there is the problem of the 
semantic correlate of a term having simple supposition. This is a 
very difficult philosophical problem, and the Scholastics were of 
varying opinions about its solution. In 27.18 Peter seems to think 
that a term with simple supposition stands for the essence (nature) 
of the object. On the other hand Ockham and his school hold that 
the semantic correlate of such a term is simple, 'the intention of the 
soul': 

27.21 A term cannot have simple or material supposition 
in every proposition but only when ... it is linked with another 
extreme which concerns an intention of the soul or an utterance 
or something written. E.g. in the proposition 'a man is running' 
the 'man' cannot have simple or material supposition, since 
'running' does not concern either an intention of the soul, nor 
an utterance nor something written. But in the proposition 
'man is a species' it can have simple supposition because 
'species' signifies an intention of the soul. 

In this and similar texts (27.22) it is of logical interest that 
Ockham and his followers were apparently trying to give an exten- 
sional interpretation even to terms having simple supposition; 
their correlates would be (concrete) intentions. 

After Buridan there were in the Middle Ages, as at the beginning 
of the 20th century, some logicians who equated simple and material 
supposition. Paul of Venice gives that information : 

27.23 Simple supposition is distinct from material and 
personal ; some say otherwise, and make no distinction between 
simple and material supposition. But (wide) it is evident that 
the subject does not suppose materially when it is said: 'the 
divine essence is inwardly communicable'. 



D. PERSONAL SUPPOSITION 

The most usual supposition of a term is personal. As Ockham 

says: 

27.24 It is also to be noticed that in whatever proposition 
it be put, a term can always have personal supposition, unless 
it be restricted to some other by the will of those who use it. 

171 



SCHOLASTIC LOGIC 

We give the definition and divisions of this kind of supposition 
according to Peter of Spain, whose text contains the essentials of 
the doctrine that remained standard till the end of the scholastic 
period. 

27.25 Personal supposition is the taking of a common term 
for its inferiors, as when it is said 'a man runs', the term 'man' 
supposes for its inferiors, viz. for Socrates and for Plato and 
so on. 

27.26 Of personal suppositions one kind is determinate, 
another confused. Determinate supposition is the taking of a 
common term put indefinitely or with the sign of particularity, 
as 'a man runs' or 'some man runs', and both are called 
'determinate' because although in both the term 'man' 
supposes for every man, whether running or not, yet they are 
true only for one man running. For it is one thing to suppose 
(for things), and another to render the proposition true for 
one of them*. But as has been said, the term 'man' supposes 
for all whether running or not, yet renders the propositions 
true only for one who is running. But it is clear that the 
supposition is determinate in both (propositions), because 
when it is said: 'An animal is Socrates, an animal is Plato, 
and so on, therefore every animal is every man', this is the 
fallacy of the form of speech (proceeding) from a number of 
determinates to one (cf. 11.19 and 27.18). And so a common 
term put indefinitely has determinate supposition, and 
similarly if it has the sign of particularity. 

27.27 But confused supposition is the taking of a common 
term for a number of things by means of the sign of univer- 
sality, as when it is said: 'every man is an animal', the term 
'man' is taken for a number by means of the sign of uni- 
versality, being taken for each of its individuals. 

Subsequent division of confused supposition into that which is 
confused by the requirements of the sign (necessitate signi) and that 
which is confused by the requirements of the thing (rei) (27.28) is 
shortly after rejected by Peter. He gives a further division of perso- 
nal supposition: 

27.29 Of personal supposition one kind is restricted, 
another extended (ampliata). 

* Reading praedictis for praedicatis. 

172 



AMPLIATION 

E. INTERPRETATION IN MODERN TERMS 

If we ask how the expression 'supposition' is to be rendered in 
modern terms, we have to admit that it cannot he. 'Supposition' 

covers numerous semiotic functions for which we now have no 
common name. Some kinds of supposition quite clearly belong to 
semantics, as in the case of both material suppositions, and personal ; 
others again, such as simple supposition and those into which 
personal supposition is subdivided, are as Moody has acutely remark- 
ed (27.30), not semantical but purely syntactical functions. 

The most notable difference between the doctrine of supposition 
and the corresponding modern theories lies in the fact that while 
contemporary logic as far as possible has one sign for one function, 
e.g. a sign for a word, another for the word's name, one for the word 
in personal, another for it in simple supposition, the Scholastics 
took equiform signs and determine their functions by establishing 
their supposition. And this brings us back to the fundamental 
difference already remarked on between the two forms of formal 
logic; scholastic logic dealt with ordinary language, contemporary 
logic develops an artificial one. 



§28. AMPLIATION, APPELLATION, ANALOGY 

Among the other properties of terms three that seem to be of parti- 
cular interest for formal logic will be illustrated with some texts, 
viz. ampliation, appellation and analogy. 



A. AMPLIATION 
Peter of Spain writes: 

28.01 Restriction is the narrowing of a common term from 
a wider (maiore) supposition to a narrower, as when it is said 
'a white man runs' the adjective 'white' restricts 'man' to 
supposing for white ones. Ampliation is the extension (exiensio) 
of a common term from a narrower supposition to a wider, as 
when it is said 'a man can be Antichrist' the term 'man' 
supposes not only for those who are now, but also for those who 
will be. Hence it is extended to future ones. I say 'of a common 
term' because a discrete term is neither restricted nor extended. 

One kind of ampliation is effected by a verb, as by the 
verb 'can', e.g. 'a man can be Antichrist'; another is effected 
by a name, e.g. 'it is possible that a man can be Antichrist' ; 
another by a participle, e.g. 'a man is able (potens) to be 
Antichrist' ; another by an adverb, e.g. 'a man is necessarily an 

173 



SCHOLASTIC LOGIC 

animal'. For (in the last) 'man' is extended not only for the 
present but also for the future. And so there follows another 
division of ampliation : one kind of ampliation being in respect 
of supposita, e.g. 'a man can be Antichrist', another with 
respect to time, e.g. 'a man is necessarily an animal', as has 
been said. 

Essentially the same doctrine but more thoroughly developed is 
found at the end of the 14th century in Albert of Saxony: 

28.02 Ampliation is the taking of a term for one or more 
things beyond what is actually the case: for that or those 
things for which the proposition indicates (denotat) that it is 
used. Certain rules are established in this respect: 

28.03 The first is this: every term having supposition in 
respect of a verb in a past tense is extended to stand for what 
was, e.g. when it is said: 'the white was black', 'the white' is 
taken in this proposition not only for what is white but for 
what was white. 

28.04 Second rule : a term having supposition in respect of 
a verb in a future tense is extended to stand for what is or 
will be. . . . 

28.05 Third rule : every term having supposition with 
respect to the verb 'can' is so extended as to stand for what is 
or can be. E.g. 'the white can be black' means that what is 
white or can be white, can be black. . . . 

28.06 Fourth rule : A term having supposition in respect of 
the verb 'is contingent' is extended to stand for what is or 
can contingently be (contingit esse). And that is Aristotle's 
opinion in the first book of the Prior {Analytics). . . . 

28.07 Fifth rule: A term subjected in a proposition in 
respect of a past participle, even though the copula of this 
proposition is a verb in the present, is extended to stand for 
what was. . . . E.g. in the proposition 'a certain man is dead' 
the subject stands for what is or has been. 

28.08 Sixth rule : In a proposition in which the copula is in 
the present, but the predicate in the future, the subject is 
extended to stand for what is or will be. E.g. 'a man is one 
who will generate'; for this proposition indicates that one 
who is or will be a man is one who will generate. 

28.09 Seventh rule: If the proposition has a copula in the 
present and a predicate that includes the verb 'can', as is the 
case with verbal names ending in '-ble' ('-ibile'), then the sub- 

174 



APPELLATION 

ject is extended to stand for what is or can be, e.g. when it is 
said: 'the man is generable'. For this is equivalent (valet) to: 
'the man can be generated' in which 'man' is extended, 
according to the third rule, to stand for what is or can be. . . . 

28.10 Eighth rule: all verbs which, although not in the 
present, have it in their nature to be able to extend to a future, 
past or possible thing as to a present one, extend the terms to 
every time, present, past and future. Such e.g. are these: 
'I understand', 'I know', 'I am aware', 'I mean (significo)' etc. 

20.11 Ninth rule: the subject of every proposition de 
necessario in the divided sense (cf. § 29, D.) is extended to 
stand for what is or can be. E.g. 'every B is necessarily A'; 
for this is equivalent to (valet dicere) 'Whatever is or can be 
B, is necessarily A'. . . . 

28.12 Tenth rule: if no ampliating term is present in a 
proposition, its subject is not extended but this proposition 
indicates that (the subject stands) only for what is. 

This text is a fine example of scholastic analysis of language. It 
introduces a notable enlargement of the doctrine of supposition, 
dividing the objects for which a term may stand into three tem- 
poral classes to which is added the class of possible objects. It 
can readily be seen that this doctrine makes an essential contri- 
bution to the problem of the so-called void class, since the expres- 
sion 'void class' receives as many different denotata as there are 
kinds of ampliation. This can be compared with the modern me- 
thods of treating the problem (cf. § 46, A and B). 

Albert's seventh, eighth and ninth rules also contain an analysis 
of modal propositions, but this subject will be considered in greater 
detail below (§ 33). 

B. APPELLATION 

Closely connected with ampliation is the so-called appellation, 
also relevant to the problem of the void class. The theory of it was 
already well developed in the 13th century, was further enlarged in 
the 14th when there were various theories different from that of the 
13th.* We cite two 13th century texts, one from Peter of Spain 
and one from Shyreswood: 

28.13 Appellation is the taking of a term for an existent 
thing. I say 'for an existent thing' since a term signifying a 
non-existent has no appellation, e.g. 'Caesar' or 'Anti-christ' 

* For information on these points and much other instruction on the doc- 
trines of supposition and appellation I am obliged to Prof. E. Moody. 

175 



SCHOLASTIC LOGIC 

etc. Appellation differs from supposition and signification in 
that appellation only concerns existents, but supposition and 
signification concern both existents and non-existents, e.g. 
'Antichrist' signifies Antichrist and supposes for Antichrist 
but does not name (appellai) him, whereas 'man' signifies man 
and naturally supposes for existent as well as non-existent 
men but only names existent ones. 

Of appellations one kind belongs to common terms such as 
'man', another to singular terms such as 'Socrates'. A singular 
term signifies, supposes and names the same thing, because it 
signifies an existent, e.g. 'Peter'. 

Further, of appellations belonging to common terms one 
kind belongs to a common term (standing) for the common 
thing itself, as when a term has simple supposition, e.g. when 
it is said : 'man is a species' or 'animal is a genus' ; and then the 
common term signifies, supposes and names the same thing, 
as 'man' signifies man in general and supposes for man in 
general and names man in general. Another kind belongs to a 
common term (standing) for its inferiors, as when a common 
term has personal supposition, e.g. when it is said : 'a man runs'. 
Then 'man' does not signify, suppose and name the same 
thing; because it signifies man in general and supposes for 
particular men and names particular existent men. 

28.14 Supposition belongs to (inest) a term in so far as it 
is under another. But appellation belongs to a term in so far 
as it is predicable of its (subordinate) things by means of the 
verb 'is'. . . . Some say therefore that the term put as subject 
supposes, and that put as predicate names. ... It should also 
be understood that the subject-term names its thing, but not 
qua subject. The predicate-term on the other hand names it 
qua predicate. 

The following from Buridan may serve as an example of 14th 
century theories:* 

28.15 First it is to be understood that a term which can 
naturally suppose for something names all that it signifies 
or consignifies unless it be limited to what it stands for. . . . 
E.g. 'white' standing for men names whiteness, and 'great' 
greatness, and 'father' the past (act of) generation and someone 

* These texts were communicated by Prof. E. Moody who also pointed out 
their great importance. He is to be thanked also for the main lines of the commen- 
tary. 

176 



ANALOGY 

else whom the father has generated, and 'the distant' names 
that from which it is distant and the space (dimensionem) 
between them by which it is made distant. . . . 

28.16 A term names what it names as being somehow 
determinant (per modum adiacenlis aliquo modo) or not 
determinant of that for which it stands or naturally can 
stand. . . . 

Thirdly it is to be held that according to the different 
positive kinds of determination of the things named - the things 
for which the term stands - there are different kinds of 
predication, such as how, how many, when, where, how one is 
related to another, etc. It is from these different kinds of 
predication that the different predicaments are taken . . . 
(cf. 11.15). 

28.17 Appellative terms name differently in respect of an 
assertoric verb in the present and in respect of a verb in the 
past and in the future, and in respect of the verb 'can' or of 
'possible'; since in respect of a verb in the present the appel- 
lative term - provided there is not ampliative term - whether 
it be put as subject or predicate, names its thing as something 
connected with it in the present, for which the term can 
naturally stand, and as connected with it in this or that man- 
ner, according to which it names. 

This is a different doctrine from that of the 13th century, and 
seems to be of the highest importance. For according to it a term 
does not name what it stands for but something related to it by, 
it would seem, any relation. Buridan says this expressly for the term 
'distant'. If A is distant from B 'distant' does not name A, but 
precisely B. That indicates a clear notion of relation-logic. Where 
we should write 'relation', Buridan has adiacentia. Especially im- 
portant is 28.16 where Buridan goes so far as to say that absolute 
terms are definable by relations, an idea corresponding to the relative 
descriptions of 47.20. Some interesting results would follow from the 
detailed working out of the basic notions of this text, e.g. a theory 
of plural quantification, but we have no knowledge of this being 
done in the Middle Ages. 

C. ANALOGY 

In the present state of research it is unfortunately impossible 
to present the scholastic theory of meaning with any hope of 
doing justice even only to its essentials. However, we shall treat of 
a further important point in this field, the theory of analogy. This 

177 



SCHOLASTIC LOGIC 

is of direct relevance to formal logic, and fairly well explored. A 
single text from Thomas Aquinas will suffice: 

28.18 Nothing can be predicated univocally of God and 
creatures; for in all univocal predication the sense (ratio) of 
the name is common to both things of which the name is 
univocally predicated . . . and yet one cannot say that what is 
predicated of God and creatures is predicated purely equivo- 
cally. ... So one must say that the name of wisdom is predi- 
cated of God's wisdom and ours neither purely univocally 
nor purely equivocally, but according to analogy, by which is 
just meant: according to a proportion. But conformity 
(convenientia) according to a proportion can be twofold, and 
so a twofold community of analogy is to be taken account of. 
For there is a conformity between the things themselves which 
are proportioned to one another in having a determinate 
distance of some other relationship (habitudinem) to one ano- 
ther, e.g. (the number) 2 to unity, 2 being the double. But we 
also sometimes take account of conformity between two things 
which are not mutually proportioned, but rather there is a 
likeness between two proportions; e.g. 6 is conformed to 4 
because as 6 is twice 3 so 4 is twice 2. The first conformity then 
is one of proportion, but the second of proportionality. So it is 
then that according to the first kind of conformity we find 
something predicated analogically of two things of which 
one has a relationship to the other, as being is predicated of 
substance and of accident owing to the relationship which 
substance and accident have (to one another), and health is 
predicated of urine and animals, since urine has some relation- 
ship * to the health of animals. But sometimes predication is 
made according to the second kind of conformity, e.g. the 
name of sight is predicated of corporeal sight and of intellect, 
because as sight is in the eye, so intellect is in the mind. 

This is about the clearest text of the many in which Thomas 
Aquinas speaks of analogy (28.19). It has been only too often 
misunderstood, but deserves fairly thorough discussion from the 
historian of logic because of its historical as well as systematic 
significance. We therefore draw attention to the following points: 

This text deals explicitly with a question of semantics - Thomas 
speaks of names - and it is noteworthy that he himself, like his best 
commentator Cajetan (28.20), almost always considers analogy as 

* Reading habitudinem for similitudinem. 

178 



ANALOGY 

'of names'. Of course he does not mean mere utterances, but mean- 
ingful words, in accordance with the scholastic usage illustrated 

previously. 

Now our text speaks of three classes of names: univocal, equivo- 
cal and analogous names. The last are intermediate between the 
two first. The class of analogous names falls into two sub-classes: 
those analogous according to a proportion, and those according to 
proportionality. Both these divisions originate with Aristotle 
(10.29 and 10.31), but the hasty indications of the Nichomachean 
Ethics are here developed into a systematic logical doctrine. 

While the thomist doctrine of the first class of analogous names 
is here only of interest as showing an attempt to formalize the rules 
of their use, the theory of the second class, i.e. of names analogous 
according to proportionality, is nothing less than a first formulation 
of the notion of isomorphy (cf. 47.41). That this is so can be seen as 
follows : 

Let us note first that according to the text an analogous name of 
the second kind always refers to a relation or relata defined by one. 
Certainly something absolute is also implied by each of the subjects 
in such an analogy, but this is precisely different in each, and in 
that respect the name is equivocal. The community of reference 
consists only in regard to certain relations. 

But it is not a matter of just one relation, rather of two similar 
ones. This is explicit in the text, only the example (6:3 = 4:2) is 
misleading since we have there an identity of two relations. That 
Thomas is not thinking of such is shown by the illustrations, first in 
the domain of creatures (sight: eye — intellect: mind), then in God 
(divine being: God - creaturely being: creature). The ruling idea is 
then of a relation of similarity between two relations. 

This relation between relations is such as to allow inference from 
what we know about one to something about the other, though at 
the same time we have the assertion : 'we cannot know what God is' 
(28.21). The apparent contradiction disappears when it is realized 
that we are dealing with isomorphy. For this does in fact allow one 
to transfer something from one relation to another, without afford- 
ing any experience of the relata. 

The use of a mathematical example is noteworthy, taken more- 
over from the only algebraic function then known. This is not only to 
be explained by the mathematical origin of the doctrine of analogy 
in Aristotle, but also perhaps by a brilliant intuition on the part of 
Aquinas who dimly guessed himself to be establishing a thesis about 
structure. In any case the text is of the utmost historical importance 
as being the first indication of a study of structure, which was to 
become a main characteristic of modern science. 



179 



§ 29. STRUCTURE AND SENSE OF PROPOSITIONS 

A. DIVISION OF PROPOSITIONS 

We give first a text of Albert of Saxony which summarizes the 
commonly received scholastic doctrine of the kinds of atomic 
(categorical) propositions : 

29.01 Of proposition some are categorical, others hypo- 
thetical. But some of the categorical are said to be hypothetical 
in signification, such as the exclusive, exceptive and redupli- 
cative propositions, and others besides. 

Then of the categorical propositions that are not equi- 
valent to the hypothetical in signification - such as 'man is an 
animal' and such-like - some are said to be assertoric (de 
inesse) or of simple inherence; others are said to be modal or 
of modified inherence. . . . 

Again of categorical propositions of simple inherence some 
have ampliative subjects, as 'a man is dead', 'Antichrist will 
exist', the others do not have ampliative subjects, as 'man is an 
animal', 'stone is a substance' etc. 

Again, of categorical propositions of simple inherence 
with ampliative subjects, some concern the present, others the 
past, others the future. . . . 

Again, of categorical propositions about the present some 
are de secundo adiacente, others de tertio adiacente. An example 
of the first : 'man exists' ; of the second : 'man is an animal'. 

Again, of categorical propositions some have a non-com- 
pound extreme (term) (de extremo incomplexo), as 'man is an 
animal'; others have a compound extreme, as 'man or ass is 
man or ass'. 

B. ANALYSIS OF PROPOSITIONS 

Here we assemble a few aspects of the scholastic analysis of 
propositions. To begin with, this text of Thomas Aquinas, followed 
by one from Ockham, about the general structure: 

29.02 In every true affirmative proposition the subject 
and predicate must signify somehow the same thing in reality 
but in different senses (diversum secundum rationem). And this 
is clear both in propositions with accidental predicate and in 
those with substantial. For it is evident that 'man' and 
'white' are identical in suppositum and differing in sense, for 
the sense of 'man' is other than the sense of 'white'. And 

180 



PROPOSITIONS 

likewise when I say: 'man is an animal', for that same thing 
which is man is truly an animal. For in one and the same 
suppositum there is both the sensible nature, after which it is 
called 'animal', and the rational nature, after which it is called 
'man'. So that in this case too the predicate and subject are 
identical as to suppositum, but differing in sense. But this is 
also found in a way in propositions in which something (idem) 
is predicated of itself, inasmuch as the intellect treats as 
suppositum (trahit ad partem suppositi) what it posits as 
subject, but treats as form inhering in the suppositum what it 
posits as predicate. Hence the adage, that predicates are 
taken formally and subjects materially (cf. 27.08). To the 
difference in sense there corresponds the plurality of predi- 
cate and subject; but the intellect signifies the real identity by 
the composition (of the two). 

We have here actually two analyses of propositions. First an 
extensional one, which seems to have become classical in later 
Scholasticism. It can be reproduced thus: the proposition 'S is P' 
is to be equated with the product of the following propositions: 
(1) there is at least one x such that both l S' and 'P' stand (suppose) 
for x, (2) there is a property / such that 'S' signifies /, (3) there is a 
property g such that 'P' signifies g, (4) both / and g belong to x. 

In the second analysis the subject is conceived as extensional, the 
predicate as intensional. The proposition 'A = A' can be interpreted : 
(1) there is an x such that 'A' stands for x, (2) there is a property / 
such that 'A' signifies /, (3) / belongs to x. This analysis is applied in 
the text to a special kind of proposition, asserting an identity, but 
can evidently be applied generally. 

Ockham gives another analysis : 

29.03 It is to be said that it is not required for the truth of 
a singular proposition, which is not equivalent to many 
propositions, that subject and predicate should be really 
identical, nor that the predicated reality should be in the 
subject, nor that it should really inhere (insit) in the subject, 
nor that it should be really, extra-mentally, united to the 
subject. E.g. it is not required for the truth of this: 'that one 
is an angel', that the common term 'angel' should be in reality 
the same as what is posited as subject, nor that it should 
really inhere in it, nor anything of that kind, - but it is 
sufficient and necessary that subject and predicate should 
suppose for the same thing. And so in this : 'this is an angel', if 
subject and predicate suppose for the same thing, the pro- 

181 



SCHOLASTIC LOGIC 

position is true. And so it is not indicated (denotatur) that this 
has angelicity, or that angelicity is in it, or anything of this 
kind, but it is indicated that this is truly an angel, not that it 
is that predicate, but that it is that for which the predicate 
supposes. 

An important text, but not readily intelligible to a modern reader. 
A possible, though not the only possible interpretation is this: it is 
sufficient and necessary for the truth of a proposition of this kind 
that the extension of subject and predicate should coincide. If that 
is so, he means to say that the predicate is not to be taken intension- 
ally, but extensionally like the subject, as in Thomas's first ana- 
lysis in 29.02. Then Ockham gives a radically extensional inter- 
pretation of propositions. 

The next text shows that what was said in 29.03 holds for other 
kinds of proposition as well: 

29.04 For the truth of such (i.e. indefinite or particular 
propositions) it suffices that the subject and predicate stand 
for the same thing, if the proposition is affirmative. 

C. ANALYSIS OF MODAL PROPOSITIONS: DICTUM AND MODUS 

In the middle of the 13th century there arose a generally accepted 
doctrine about the structure of modal propositions. It is to be found 
in Albert the Great (29.05), Shyreswood (29.06), Peter of Spain (29.07), 
and in the Summa Toiius Logicae (29.08). On account of its charac- 
teristic formalism we quote a youthful opusculum of Thomas 
Aquinas: 

29.09 Since the modal proposition gets its name from 
'modus', to know what a modal proposition is we must know 
what a modus is. Now a modus is a determination of something 
effected by a nominal adjective determining a substantive, 
e.g. 'white man', or by an adverb determining a verb. But it 
is to be known that modes are threefold, some determining 
the subject of a proposition, as 'a white man runs', some deter- 
mining the predicate, as 'Socrates is a white man' or 'Socrates 
runs quickly', some determining the composition of the 
predicate with the subject, as 'that Socrates is running is 
impossible', and it is from this alone that a proposition is said 
to be modal. Other propositions, which are not modal, are said 
to be assertoric (de inesse). 

The modes which determine the composition are six: 'true', 
'false', 'necessary', 'possible', 'impossible' and 'contingent'. 

182 



PROPOSITIONS 

But 'true' and 'false' add nothing to the signification of asser- 
toric propositions; for there is the same significance in 
'Socrates runs' and it is true that Socrates runs' (on the one 
hand), and in 'Socrates is not running' and 'it is false that 
Socrates is running' (on the other). This does not happen with 
the other four modes, because there is not the same significance 
in 'Socrates runs' and 'that Socrates runs is impossible (or 
necessary)'. So we leave 'true' and 'false' out of consideration 
and attend to the other four. 

Now because the predicate determines the subject and not 
conversely, for a proposition to be modal the four modes 
aforesaid must be predicated and the verb indicating com- 
position must be put as subject. This is done if an infinitive is 
taken in place of the indicative verb in the proposition, 
and an accusative in place of the nominative. And it (the 
accusative and infinitive clause) is called 'dictum', e.g. of the 
proposition 'Socrates runs' the dictum is 'that Socrates runs' 
(Socratem currere). When then the dictum is posited as subject 
and a mode as predicate, the proposition is modal, e.g. 'that 
Socrates runs is possible'. But if it be converted it will be 
assertoric, e.g. 'the possible is that Socrates runs'. 

Of modal propositions one kind concerns the dictum, 
another concerns things. A modal (proposition) concerning 
the dictum is one in which the whole dictum is subjected and 
the mode predicated, e.g. 'that Socrates runs is possible'. A 
modal (proposition) concerning things is one in which the 
mode interrupts the dictum, e.g. 'for Socrates running is 
possible' (Socratem possibile est currere). But it is to be known 
that all modals concerning the dictum are singular, the mode 
being posited as inherent in this or that as in some singular 
thing. But . . . modals concerning things are judged to be 
universal or singular or indefinite according to the subject of 
the dictum, as is the case with assertoric propositions. So 
that 'for all men, running is possible' is universal, and so 
with the rest. It should further be known that modal pro- 
positions are said to be affirmative or negative according 
to the affirmation or negation of the mode, not according to 
the affirmation or negation of the dictum. So that . . . this 
modal 'that Socrates runs is possible' is affirmative, while 
'that Socrates runs is not possible' is negative. 

There are two notable points in this text. First there is the very 
thorough formalism, the modal proposition being classified accord- 

183 



SCHOLASTIC LOGIC 

ing to the place which the mode has in it. Then there is the explicit 
distinction of the two structures, one of which Aristotle made the 
basis of his modal logic (§ 15, B), the other of which Theophrastus 
adopted (§ 17, B). The modals de re correspond to the Aristotelian 
structure, in which the mode does not determine the 'composition' 
or, as we should say, the proposition as a whole, but 'the predicate'. 
Taking the proposition A is possibly B J as de re, we could analyze 
it thus: 

if x is A, then x is possibly B. 
But the modals de dido have the Theophrastan structure, according 
to which the fore-going proposition will be taken as de dido and can 
be interpreted: 

that A is B is possible. 



D. COMPOSITE AND DIVIDED SENSES 

Closely connected with that doctrine, classical in Scholasticism, 
is that of the composite and divided senses of propositions. It was 
developed out of the Aristotelian theory of the fallacies of division 
and composition (11.22f.), and partly corresponds to the foregoing 
analysis of modal propositions (29.09), but extends to other kinds 
as well. It seems to have secured a quite central place in later 
scholastic logic. We cite first a text of Peter of Spain: 

29.10 There are two kinds of composition. The first kind 
arises from the fact that some dictum can suppose for itself 
or a part of itself, e.g. 'that he who is sitting walks is possible'. 
For if the dictum 'that he who is sitting walks' is wholly 
subjected to the predicate 'possible', then the proposition is 
false and composite, for then opposed activities, sitting and 
walking, are included in the subject, and the sense is: 'he 
who is sitting is walking'. But if the dictum supposes for a part 
of the dictum, then the proposition is true and divided, and 
the sense is: 'he who is sitting has the power of walking'. To 
be distinguished in the same fashion is : 'that he who is not 
writing is writing is impossible'. For this dictum 'that he 
who is not writing is writing' is subjected to the predicate 
'impossible',* but sometimes as a whole, sometimes in respect 
of a part of itself. And similarly: 'that a white thing is black 
is possible'. And it is to be known that expressions of this 
kind are commonly said to be de re or de dicio. 

Reading impossibile for possibile. 

184 



PROPOSITIONS 

A twofold terminology can be seen here ; the distinction composita- 
divisa corresponds to de dido - de re. Peter also introduces the notion 
of supposition, while Thomas (29.09) proceeds wholly syntactically. 
Thomas has yet other expressions for the same idea: 

29.11 Further (it is objected), if everything is known by 
God as seen in the present, it will be necessary that what God 
knows, is, as it is necessary that Socrates sits given that he is 
seen to be sitting. But this is not necessary absolutely, or as is 
said by some, by necessity of the consequent: rather condi- 
tionally, or by necessity of consequence. For this conditional 
is necessary: If he is seen to be sitting, he sits. Whence also, if 
the conditional is turned into a categorical, so that it is said: 
what is seen to be sitting, necessarily sits, evidently if this is 
understood as de dido and composite, it is true ; but understood 
as de re and divided, it is false. And so in these and all similar 
cases . . . people are deceived in respect of composition and 
division. 

This gives us the two following series of expressions, correspond- 
ing member to member (the word propositio being understood with 
each) : de dido, composita, necessaria necessitate consequentiae, 
necessaria sub conditione - de re, divisa, necessaria necessitate conse- 
queniis, necessaria absolute. 

Paul of Venice gives a peculiar variant of the doctrine of de dido 
and de re: 

29.12 Some say that always when the mode simply precedes 
or follows the expression with the infinitive, then the sense is 
definitely called 'composite' in every case, e.g. 'it is possible 
that Socrates runs', 'that Socrates runs (Socratem currere) is 
possible'. But when the mode occupies a place in the middle the 
sense is called 'divided', e.g. 'for Socrates it is possible to 
run'. Others on the other hand say that when the mode 
simply precedes, the sense is composite, as previously, but 
when it occupies a middle place or comes at the end, then the 
sense is divided, e.g. 'of A I know that it is true', 'that A is 
true is known by me'. And so with others similar. 

But though these ways of speaking enjoy probability, yet 
they are not wholly true. ... So I say otherwise, taking a 
position intermediate between them: when the mode simply 
precedes a categorical or hypothetical dictum, it effects the 
composite sense; and when it occurs between the verb and 
the first extreme, it is taken in the divided sense; but when it 

185 



SCHOLASTIC LOGIC 

follows at the end, it can be taken in the composite or the 
divided sense. 

This seemingly purely grammatical text is yet not without 
interest as showing how scholastic logic at the end of the 14th 
century was wholly bent on grasping the laws of everyday language. 
We find no essentially new range of problems in Paul beyond those 
of Thomas and Peter. 

There is yet another interpretation of the composite and divided 
senses, first found in Peter, in a text which seems to adumbrate all 
the associated problems: 

29.13 (The fallacy of) division is a false division of things 
that should be compounded. There are two kinds of division. 
The first arises from the fact that a conjunction can conjoin 
either terms or propositions, e.g. . . . Tive is even or odd'. 
Similarly: 'every animal is rational or irrational'. For if this 
conjunctive particle 'or' divides one proposition from another, 
it is false, and its sense is: 'every animal is rational or every 
animal is irrational'. If it disjoins one term from another, 
then it is true and its sense is: 'every animal is rational or 
irrational', in which the whole disjunctive complex is predi- 
cated. Similarly: 'every animal is healthy or sick', 'every 
number is even or odd'. 

A more exact formulation of the same thought occurs in Burleigh : 

29.14 'Every animal is rational or irrational.' The proof is 
inductive. The disproof runs: every animal is rational or 
irrational, but not every animal is rational, therefore every 
animal is irrational. The conclusion is false, the minor is not 
(false), therefore the major is. The consequence is evident 
from the locus of opposites (16.18). 

Solution. The first (proposition) is multiple, according to 
composition and division. In the sense of composition it is 
true, in the sense of division it is false. Induction does not 
hold in the sense of division, because in the sense of division 
there is not a categorical proposition but a hypothetical of 
universal quantity; and thus the answer to the proof is 
clear. 

To the disproof, I say the consequence does not hold in the 
sense of composition, nor is there room for an argument from 
the locus of opposites, for the locus of opposites is when one 
argues from a disjunctive and the negation of one part, to 

186 



PROPOSITIONS 

the other part; but in the sense of composition this is not a 
disjunctive but a categorical proposition. 

The form of propositions of this kind in the composite sense could 

be expressed with variables thus: 

(1) for every x: x is A or a; is B. 
In the divided sense the same proposition could be interpreted : 

(2) Tor every x: x is A, or for every x: x is B. 
If this interpretation is correct we have here an important theorem 
about the distribution of quantifiers. Yet Burleigh does Dot seem 
to have been thinking quite of (1), but rather of 

(V) for every x: x is (A or B). 

E. REFERENCE OF PROPOSITIONS 

Finally we give a text in which the chief scholastic theories about 
the semantic correlate of propositions are listed. It is taken from 
Paul of Venice. 

29.15 About the essence of the proposition . . . there are 
many opinions. 

The first is that the significate of a true proposition is a 
circumstance (modus) of the thing and not the thing itself. . . . 

29.16 The second opinion is that the significate of a true 
proposition is a composition of the mind (mentis) or of the 
intellect which compounds or divides. . . . 

29.17 The third opinion, commonly received among the 
doctors of my (Augustinian) Order, in particular by Master 
Gregory of Rimini, is that the significate of a proposition is 
whatever in any way exists as a signifiable complex. And 
when it is asked whether such a signifiable is something or 
nothing, he answers that the name 'something' and its syno- 
nyms 'thing' and 'being' can be understood in three ways. 
(1) First in the widest sense, according to which everything 
signifiable, with or without complexity, truly or falsely, is 
called 'thing ' and 'something'. ... (2) In a second way these 
(names) are taken for whatever is signifiable, with or without 
complexity, but truly, ... (3) In a third way the aforesaid 
names are taken in such wise that they signify some existent 
essence or entity, and in this way, what does not exist is 
called 'nothing' .... So this opinion says that the significate 
of a proposition is something, if one takes the afore-mentioned 
terms in the first or second way. . . . 

29.18 The fourth opinion posits some theses. (1) The first 
is this : that no thing is the adequate or total significate of a 

187 



SCHOLASTIC LOGIC 

mental proposition properly so called; since every such 
(proposition) signifies a variety of mutually distinct things, 
by reason of its parts to which it is equivalent in its signifying. 
And this is evident to everyone who examines the matter. 
Hence there is no total or adequate significate of such a 
proposition. 

(2) The second thesis: whatever is signified by a mental 
proposition properly so called according to its total significa- 
tion is also signified by any of its parts. . . . 

(3) Third thesis: no dictum corresponding to a mental 
proposition properly so called, e.g. an expression in the 
infinitive mood taken as significant, supposes for any thing. 
For instance, if the dictum, i.e. the expression in the infinitive 
mood, 'that man is an animal', corresponding to the proposi- 
tion 'man is an animal', is taken materially it stands for some 
thing, namely for the proposition to which it corresponds; 
but if it be taken significatively, i.e. personally, then according 
to the fourth opinion it stands for no thing. This is evident, 
since such an expression, so taken, signifies a number of 
things, viz. all those signified by the corresponding proposition, 
and so there would be no reason for it to suppose for one of 
its significates rather than another; hence (it supposes) 
either for each or for none. But nobody would say for each, 
since the expression 'that man is an animal' would signify 
an ass or suppose for an ass. Therefore for none. And what is 
said of that instance, holds for any other. 

The four opinions there enumerated could be summed up thus 
in modern terminology; a proposition has for its semantical correlate 
(1) a real fact, (2) a psychical act ( 3) an objective content (the Stoic 
lekton), (4) nothing at all beyond what its parts already signify. 
In the 15th century there were very complicated and sharp disputes 
about this problem. But as it lies on the border-line of pure logic 
we shall omit consideration of them here (29.19). 



188 



II. PROPOSITIONAL LOGIC 



§ 30. NOTION AND DIVISION OF CONSEQUENCES 



A. HISTORICAL SURVEY 

The theory of consequences is one of the most interesting scholastic 
doctrines. Essentially it is a development of Stoic prepositional 
logic, though so far as is known it was constructed entirely anew, 
not in connection with the Stoic logoi (§ 21 ) but with certain passages 
of the Hermeneia and, above all, the Topics. All the same, fragments 
of the Stoic propositional logic did influence the Scholastics, mostly 
through the mediation of Boethius, though for a few we must 
suppose some other sources, as e.g. for the 'dog-syllogism' (22.19;, 
which is found in Thomas Aquinas (30.01). But that these fragments 
were not the starting point is clear from the fact that, at least to 
begin with, they are not cited in the treatise on consequences, but 
in another on hypothetical syllogisms. 

The name ' consequential' is Boethius's translation of Aristotle's 
axoXoi>07](7!.<; which occurs frequently in the Hermeneia (30.02) but 
not in any exact technical sense, rather for following quite in general. 
The word has the same sense in Abelard, though limited to logical 
relationships between terms (30.03),* and to some extent also in 
Kilwardby * * (30.04) and Peter of Spain (30.05) E.g. in the latter we 
read of a consequenlia esseniiae (30.06). 

In Ockham, on the other hand, and his successors the word 
has a sharply defined technical meaning, and signifies a relation of 
consequence between two propositions. 

The following text from Kilwardby may serve as a good example 
of the earlier stage: 

30.04 He (Aristotle) also says that something is a consequent 
(of something else) in part, and yet whatever follows from A 
follows from all that is contained under A, since what follows 
on the consequent follows on the antecedent, and so every 
consequent follows on the whole antecedent. . . . 

It is to be answered to this, that (Aristotle) in this whole 
treatment (Prior Analytics) takes 'consequent' for the pre- 
dicate and 'antecedent' for the subject. . . . 

A further passage from Kilwardby is extremely instructive about 
this relationship: 



Vide Translator's Preface, B. 

Cited from transcriptions of two Oxford MSS (30.04) made by the translator. 

189 



SCHOLASTIC LOGIC 

30.07 Consequence is twofold, viz. essential or natural, 
as when a consequent is naturally understood in its antecedent, 
and accidental consequence. Of the latter kind is the conse- 
quence according to which we say that the necessary follows 
on anything. . . . 

That shows that for Kilwardby a 'natural' and 'essential' conse- 
quence is only present when it is a matter of connection between 
terms. Thus for him the proposition 'every man runs, therefore 
there is a man who runs' would be natural, since 'each man' is 
'naturally' included in 'every man'. In other words it is for him 
always based on term-logical relationships. He does also recognize 
purely propositional consequences such as the one he states: 'the 
necessary follows on anything', but these he considers only 'acciden- 
tal' and of an inferior kind. 

This opinion of Kilwardby's is of interest as showing that the 
Scholastics did not take the abstract propositional logic of the Stoics 
as their starting point, but the term-logic of Aristotle. Yet before 
very long they built on that basis a technically excellent pure logic 
of propositions that to the best of our knowledge was superior to 
that of the Megarians and Stoics. 

Since the paper of Lukasiewicz (30.08) more works have been 
devoted to this propositional logic than to any other scholastic 
logical doctrine, and it is better known than most others (30.09). Yet 
we are still far from having a complete knowledge of it. We cannot 
here enumerate all the scholastic consequences that have been 
investigated in the 20th century, but must limit ourselves to texts 
defining the notion of consequentia, and then (§ 31) give a few examp- 
les. 

B. DEFINITION OF CONSEQUENCE 

Pseudo-Scotus gives the following definition of consequence: 

30.10 A consequence is a hypothetical proposition composed 
of an antecedent and consequent by means of a conditional 
connective or one expressing a reason (rationalis) which 
signifies that if they, viz. the antecedent and consequent, 
are formed simultaneously, it is impossible that the ante- 
cedent be true and the consequent false. 

Here a consequence is conceived as a proposition in almost word 
for word agreement with the Stoic definition (19.15) with only two 
considerable differences: (1) 'proposilio' means, not the lekton, but 
the thought, written and spoken proposition (cf. 26.03) ; (2) the 
consequence corresponds to the compound and inferential sentences 
of the Stoics (19.15). Implication is defined in the Diodorean way 

190 



CONSEQUENCES 

(20.08), though the time-variables might be thought to be missing; 
but comparison with the definition of consequence ut nunc (30.12, 
cf. 30.16) shows that Diodorus's idea of 'for all times' is basic for 
the Scholastics too. They conducted a complicated discussion which 
shows that the range of problems considered was much wider than 
might be expected from what we have said here (cf. 30.17 f.). 

A noteworthy exception to the premise that a consequence is a 
proposition* is found in Burleigh: 

30.11 It is also to be noted that the (contradictory) opposite 
of the antecedent does not follow from the opposite of the 
consequent in every valid consequence, but only in non- 
syllogistic consequences. For in syllogistic consequences the 
antecedent has no opposite, because a syllogistic antecedent 
is an unconnected plurality of propositions (propositio plures 
inconiuncle) and because such an antecedent has no opposite 
at all, it not being a proposition that is either simply or 
conjunctively one. But in a syllogistic consequence the 
opposite of one premiss follows from the opposite of the 
conclusion with the other premiss. And if from the opposite of 
the conclusion with one or other of the premisses there 
follows the opposite of the remaining premiss, then the original 
syllogism was valid. For that is how the Philosopher proves 
his syllogisms, viz. arguing from the opposite of the con- 
clusion with one of the premisses, as can be seen in the first 
book of the Prior Analytics. 

'Propositio plures inconiuncle' means here, as usually among the 
Scholastics (cf. 35.45) not a compound proposition, not^therefore a 
product of propositions, but a number of juxtaposed propositions. 
It follows that syllogisms, and so 'syllogistic consequences' are 
not propositions, and further that consequences were not always 
thought of as conditional propositions. 



C. DIVISION OF CONSEQUENCES 

Here again we begin with Pseudo-Scotus : 

30.12 Consequences are divided thus: some are material, 
others are formal. A formal consequence is one which holds 
in all terms, given similar mutual arrangement (disposilio) 
and form of the terms. ... A material consequence is one 

* Prof. L. Minio-Paluello tells us in connection with Cod. Orleans '266, fol. 78 
that this was already debated in the middle of the 12th century. 

191 



SCHOLASTIC LOGIC 

which does not hold in all terms given similar mutual arrange- 
ment and form so that the only variation is in the terms 
themselves. And such a consequence is twofold : one is simply 
true, the other true for the present (ut nunc). A simply true 
consequence is one reducible to a formal consequence by the 
addition of a necessary proposition. A correct material 
consequence true for the present is one which is reducible to 
a formal consequence by the addition of a true contingent 
proposition. 

So there are three kinds of consequence: (1) formal, (2) simple 
material, (3) material ut nunc. The last two are reduced to the first, 
but by means of different kinds of proposition. For (2) there is requir- 
ed a necessary, and so always true, proposition, for (3) one must 
use a contingent proposition, one which is therefore true only at a 
certain time. An example of the reduction of (2) to (1) is: 'A man 
runs, therefore an animal runs' is reduced to a consequence of kind 
(1) by means of the proposition 'every man is an animal' when it is 
said: 'every man is an animal, a man runs, therefore an animal 
runs'. The newly introduced proposition is necessary, and so always 
true, hence the consequence reduced by its means to (1) is 'simply' 
valid, valid for all time. 

Another definition is to be found in Ockham, along with a further 
division of formal consequence: 

30.13 Of consequences, one kind is formal, another material. 
Formal consequence is twofold, since one holds by an extrinsic 
medium concerning the form of the proposition, such as 
these rules: 'from an exclusive to a universal (proposition) 
with the terms interchanged is a correct consequence', 'from 
a necessary major and an assertoric minor (premiss) there 
follows a necessary (conclusion)' etc. The other kind holds 
directly through an intrinsic medium and indirectly through 
an extrinsic one concerning the general conditions of the 
proposition, not its truth, falsity, necessity or impossibility. 
Of this kind is the following: 'Socrates does not run, therefore 
some man does not run'. The consequence is called 'material' 
since it holds precisely in virtue of the terms, not in virtue 
of some extrinsic medium concerning the general conditions 
of the proposition. Such are the following: 'If a man runs, 
God exists', 'man is an ass, therefore God does not exist' etc. 

This text is most important, since Ockham here introduces a 
doctrine analogous to that of Whitehead and Russell in their distinc- 
tion of formal and material implication (44.11 ff.), analogous only, 

192 



CONSEQUENCES 

because the basic idea of implication is here Diodorean (20.08) 
instead of Philonian (20.07). Formal implications in this sense are 
further divided into two classes according as they hold in virtue of 
their component symbols or other propositions of the system. 

These ideas are defined with some accuracy in a text of Albert of 
Saxony: 

30.14 Of consequences, one kind is formal, another 
material. That is said to be a formal consequence to which 
every proposition which, if it were to be formed, would be a 
valid consequence, is similar in form, e.g. l b is a, therefore 
some a is b\ But a material consequence is one such that not 
every proposition similar in form to it is a valid consequence, 
or, as is commonly said, which does not hold in all terms when 
the form is kept the same; e.g. 'a man runs, therefore an 
animal runs'. But in these (other) terms the consequence is 
not valid: 'a man runs, therefore a log runs'. 

We may compare 26.11 f. with this text, so far as concerns the 
notion of logical form, and indeed the former follows immediately 
on the latter. 

Reverting to the distinction of simple and ut nunc consequences 
(30.12) with a similar reference to time-variables as in Diodorus 
Cronus (20.08, cf. 19.23), Burleigh formulates this idea explicitly 
and accurately: 

30.15 Of consequences, some are simple, some ut nunc. 
Simple are those which hold for every time, as: 'a man runs, 
therefore an animal runs'. Consequences ut nunc hold for a 
determinate time and not always, as: 'every man runs, 
therefore Socrates runs' ; for that consequence does not hold 
always, but only so long as there is a man Socrates. 

The first rule of consequence is this : in every valid simple 
consequence the antecedent cannot be true without the con- 
sequent. And so, if in any possible given case the antecedent 
could be true without the consequent, the consequence 
would not be valid. But in a consequence ut nunc, the ante- 
cedent ut nunc, i.e. for the (given) time for which the conse- 
quence holds, cannot be true without the consequent. 

Buridan has a text on this subject, in which occurs the new idea 
of consequence for such and such a time (ut tunc). 

30.16 Of material consequences some are said to be 
consequences simply, since they are consequences without 
qualification, it being impossible for their antecedents to be 
true without their consequents. . . . And others are said to be 

193 



SCHOLASTIC LOGIC 

consequences ut nunc, since they are not valid without 
qualification, it being possible for their antecedents to be true 
without their consequents. However, they are valid ut nunc, 
since things being exactly the same as they now are, it is 
impossible for the antecedent to be true without the conse- 
quent. And people often use these consequences in ordinary 
language (utuntur saepe vulgares), as when we say: 'the white 
Cardinal has been elected Pope' and conclude: 'therefore a 
Master in Theology has been elected Pope' ; and as when I 
say: 'I see such and such a man' . . . you conclude 'therefore 
you certainly see a false man'. But this consequence is reduced 
to a formal one by the addition of a true, but not necessary, 
proposition, or of several true and not necessary ones, as in 
the examples given, since the white Cardinal is a Master in 
Theology and since such and such a man is a false man. In 
that way the following is a valid consequence: under the 
hypothesis that there are no men but Socrates, Plato and 
Robert, 'Socrates runs, Plato runs and Robert runs; therefore 
every man runs'. For this consequence is perfected by this 
true (proposition): 'every man is Socrates, Plato or Robert'. 
And it is to be known that to this kind of consequences ut 
nunc belong permissive consequences, e.g. 'Plato says to 
Socrates: if you come to me I will give you a horse'. The 
proposition may be a genuine consequence, or it may be a 
false proposition and no consequence, since ( 1 ) if the antecedent 
is impossible, viz. because Socrates cannot come to Plato, 
then the consequence is simply speaking a genuine conse- 
quence, because from the impossible anything follows as will 
be said below. But if (2) the antecedent is false but not 
impossible, then the consequence is valid ut nunc, because 
from whatever is false anything follows, as will be said later, 
provided, however, that we restrict the name 'consequence 
ut nunc' to consequences ut tunc, whether concerning the 
past, future, or any other determinate time. But if the ante- 
cedent is true, so that Socrates will come to Plato, then 
perhaps we should say that it is still a genuine consequence 
because it can be made formal by the apposition* of true 
(propositions), when one knows whatever Plato wills to do in 
the future, that his wish will persist and that he will be able 
to carry it out; and when all circumstances are taken account 

* Reading appositas for oppositas. 

194 



CONSEQUENCES 

of according to which he wills it, and he suffers no hindrance, 
so that he will be able to and will do what and when he wills; 
if you then modify this proposition so that it is true according 
to the ninth book of the Metaphysics, i.e. 'Plato wills to give 
Socrates a horse when he comes to him; therefore Plato will 
give Socrates a horse'. If then these propositions about Plato's 
will and power are true, then Plato uttered a genuine conse- 
quence ui nunc to Socrates, but if they are not true he told 
Socrates a lie. 

D. MEANING OF IMPLICATION 

If the rooks and the crows cawed about the meaning of implication 
in the 2nd century b. c. (20.06), this occupation was surely intensi- 
fied in the 15th century. For while the Megarian-Stoic school has 
bequeathed to us only four interpretations, Paul of Venice tells us 
of ten. Not all his definitions are comprehensible to us today, but 
perhaps this is due to textual corruptions. However, for the sake of 
completeness we give the whole list. 

30.17 Some have said that for the truth of a conditional 
is required that the antecedent cannot be true without the 
consequent. . . . 

Others have said that for the truth of a conditional it is 
not required that the antecedent cannot be true without 
the consequent in the divided sense, but it is required that it 
is not possible for the antecedent to be true without the 
consequent being true. 

Thirdly people have said that for the truth of a conditional 
it is required that it is not possible that the antecedent of 
that consequence be true unless the consequent be true . . . 

Fourthly people have said that for the truth of a conditional 
it is required that it is not possible that the antecedent be 
true while the consequent of that same antecedent is false 
without a fresh interpretation (impositio) .... 

Fifthly people say that for the truth of a conditional it is 
required that if things are (ita est) as is signifiable by the 
antecedent, necessarily things are as is signifiable by the 
consequent. . . . 

Sixthly people say that for the truth of a conditional it is 
required that it be not possible that things should be so and 
not so, referring to the significates of the antecedent and of 
the consequent* of that conditional. . . . 

* Omitting oppositi. 

195 



SCHOLASTIC LOGIC 

Seventhly people say that for the truth of a conditional it 
is required that it is not possible for things to be so and not so, 
referring to the adequate significates of the antecedent and 
the consequent. . . . 

Eighthly people say that for the truth of a conditional it is 
required that the consequent be understood in the antece- 
dent. . . . 

Ninthly people say that for the truth of a condition it is 
required that the adequate significate of the consequent be 
understood in the antecedent. 

Tenthly people say that for the truth of a conditional it is 
required that the opposite of the consequent be incompatible 
with the antecedent. . . . 

For the distinction between the first two of those definitions the 
following text of Buridan is instructive. * 

30.18 Then there is the rule . . . , that the consequence is 
valid when it is impossible that things are as signified by the 
antecedent without their being as is signified by the conse- 
quent. And this rule can be understood in two ways. 

In one way so that it would be a proposition concerning 
impossibility in the composite sense (the way in which it is 
usually intended) and the meaning then is that a consequence 
is valid when the following is impossible: 'If it is formed, 
things are as is signified by the antecedent, and are not as 
is signified by the consequent.' But this rule is invalid, since 
it justifies the fallacy: 'No proposition is negative, therefore 
some propositions are negative.' 

In the other way, so that it would be a proposition concern- 
ing impossibility in the divided sense, so that the meaning is: 
a consequence is valid when whatever is stated in the ante- 
cedent cannot possibly be so without whatever is stated 
in the consequent being so. And it is clear that this rule would 
not prove the fallacy true; for whatever 'no proposition is 
negative' states, is possibly so, although things are not as the 
other (proposition of the fallacy) states; for if they were, 
affirmatives would persist but all negatives would be annihi- 
lated. 



* This text was kindly communicated by Prof. E. Moody. 

196 



CONSEQUENCES 
E. DISJUNCTION 

The notion of implication is closely connected with that of dis- 
junction. Hence two characteristic texts are in place to illustrate 
the problems connected with the latter. Peter of Spain writes: 

30.19 For the truth of a disjunctive (proposition) it is 
required that one part be true, as 'man is an animal or crow is 
a stone', and it is allowed that both of its parts be true, but 
not so properly, as 'man is an animal or horses can whinny'. 
For its falsity it is required that both of its parts be false, as 
'man is not an animal or horse is a stone'. 

Peter's idea of disjunction is evidently rather hazy, for he wavers 
between the exclusive (20.14) and the non-exclusive (20.18) dis- 
junction, describing the latter as 'less proper' though at the same 
time determining falsity in a way suitable to it alone. Which of the 
two is 'proper' must have been debated even in the 14th century, 
as can be seen from Burleigh's fine text: 

30.20 Some say that for the truth of a disjunctive it is 
always required that one part be false, because if both parts 
were false it would not be a true disjunctive; for disjunction 
does not allow those things which it disjoins to be together, 
as Boethius says. But I do not like that. Indeed I say that 
if both parts of a disjunctive are true, the whole disjunctive 
is true. And I prove it thus. If both parts of a disjunctive are 
true, one part is true; and if one part is true, the disjunctive 
is true. Therefore (arguing) from the first to the last: if both 
parts of a disjunctive are true, the disjunctive is true. 

Further, a disjunctive follows from each of its parts, but it 
is an infallible rule that if the antecedent is true, the conse- 
quent is true; therefore if each part is true the disjunctive is 
true. 

I say therefore, that for the truth of a disjunctive it is not 
required that one part be false. 

Burleigh therefore definitely sides with those who understand 
disjunction as non-exclusive. Also to be remarked in this text are 
the two propositional consequences formulated with exemplary 
exactness: 

30.201 If A and B, then A. 

30.202 If A, then A or B. 

197 



§31. PROPOSITIONAL CONSEQUENCES 

The Scholastics made no explicit difference between conse- 
quences pertaining to propositional and to term-logic. Yet they 
usually, at least after Ockham, dealt with the former first. It is 
convenient in this connection to quote a text from Paul of Venice in 
which he collects the terminology used of so-called hypothetical 
propositions. After that we give three series of texts, one from 
Kilwardby (first half of the 13th century), one from Albert of Saxony 
(second half of the 13th century) and the third from Paul of Venice 
(first half of the 15th century). To those we add some texts from 
Buridan about consequences ut nunc. We cannot claim to survey 
even the essentials of scholastic propositional logic, for this is as 
yet too little explored. The texts cited serve only as examples of 
the problems considered and the methods applied. 

A. HYPOTHETICAL PROPOSITIONS 

31.01 Some posit five kinds of hypotheticals, some six, 
others seven, others ten, others fourteen etc. But leaving all 
those aside, I say that there are three and no more kinds of 
hypotheticals that do not coincide in significance, viz. the 
copulative, disjunctive, and conditional to which the rational 
is to be counted equivalent. For I do not see that the temporal, 
local and causal are hypothetical, still less those formed and 
constituted by other adverbial and connective particles. These 
are only hypothetical by similitude, e.g. 'I have written as you 
wanted', 'Michael answers as I tell him'. Similarly the com- 
parative, e.g. 'Socrates is as good as Plato', 'Socrates is whiter 
than Plato'. Again, the relative, e.g. 'I see a man such as 
you see'. . . . Similarly the inhibitive, e.g. 'Socrates takes 
care than no-one confute him'. Again the elective, e.g. 'it is 
better to concede that your reply is bad than to concede 
something worse'. Similarly the subjunctive, e.g. 'I saw to it 
that you answered well'. Similarly the expletive, e.g. 'you 
may be moving but you are not running!' Thus by taking* 
the other particles in turn one can form a very great number 
of (pretended) hypotheticals. 

B. KILWARDBY 

We take a first series of consequences from Kilwardby's com- 
mentary on the Prior Analytics of Aristotle. Kilwardby does not 

* Reading discurrendo for distribuendo. 

198 



CONSEQUENCES 

always distinguish very clearly between propositional and term- 
relationships (cf. 30.04), so that 'antecedent' and 'consequent' must 
sometimes be understood as referring to the subject and predicate 
of universal affirmative propositions. 

31.02 What is understood in some thing or things, follows 
from it or from them by a necessary and natural consequence ; 
and so of necessity if one of a pair of opposites is repugnant to 
the premisses (of a syllogism) the other follows from them. 

31.03 If one of the opposites does not follow, the other 
can stand. 

31.04 If one of the opposites stands, the other cannot. 

31.05 What does not follow from the antecedent does 
not follow from the consequent. 

31.06 What follows from the consequent follows from the 
antecedent. 

31.07 What is compossible with one of two equivalents 
(convertib ilium) is so with the other. 

31.08 It is to be said that a negation can be negated, and 
so there is a negation of negation, but this second negation 
is really an affirmation, though accidentally (secundum quid) 
and vocally a negation. For a negation which supervenes on 
a negation destroys it, and in destroying it posits an affirma- 
tion. 

31.09 If there necessarily follows from 'A is white' l B is 
large', then from the denial (destrudio) of the consequent: if 
B is not large, A is not white. 

31.10 A disjunctive follows from each of its parts, and by a 
natural consequence; for it follows: if you sit, then you sit 
or you do not sit. 

31.11 If the antecedent is contingent or possible, so is the 
consequent. 

31.12 It is not necessary that what follows from the ante- 
cedent follows from the consequent. 



C. ALBERT OF SAXONY 

Secondly we give a series of texts from the Peruiilis Logica of 
Albert of Saxony in which the doctrine of consequences can be seen 
in a highly developed state. Albert is here so closely dependent on 
Buridan that he often simply copies him. But there is much that he 
formulates more clearly, and the available text of Buridan is not so 
good as that of the Peruiilis Logica. Buridan himself is not the 

199 



SCHOLASTIC LOGIC 

original author of his doctrine of consequences; much of it comes 
from Ockham, and some even from Peter of Spain. 

As in this whole section, the contemporary range of problems is 
only barely illustrated. 

Albert's definitions of antecedent and consequent deserve to be 
quoted first: 

31.13 That proposition is said to be antecedent to another 
which is so related to it that it is impossible that things be 
as is signifiable by it without their being as is in any way 
signifiable by the other, keeping fixed the use (impositio) of 
the terms. 

Like all Scholastics of the 14th century and after, Albert makes a 
clear distinction between a rule of consequence and the consequence 
itself. A rule is a metalogical (more exactly a meta-metalogical) 
description of the form of a valid consequence. The consequence 
itself is a proposition having this form. That generally holds good; 
but some of Albert's rules are conceived as propositional forms like 
the Stoic inference-schemata (21.22) - cf. the fifth (31.18) - only 
with this difference, that the variables are here evidently metalo- 
gical, i.e. to be substituted with names of propositions, not with 
propositions themselves as is the case with the Stoic formulae. 

31.14 The first (rule of simple consequence) is this: from an 
impossible proposition every other follows. Proof: from the 
nominal definitions of antecedent and consequent given in the 
first chapter. For if a proposition is impossible, it is impossible 
that things are as it indicates, and are not as any other 
indicates ; therefore the impossible proposition is antecedent to 
every other proposition, and hence every proposition follows 
from an impossible one. This it is which is usually expressed: 
anything follows from the impossible. And so it follows: man 
is an ass, therefore a man runs; since the antecedent being 
impossible, if things are not as the consequent indicates, it is 
impossible that they should be as the antecedent indi- 
cates. 

31.15 Second rule: A necessary proposition follows from 
any proposition. This is again proved by the nominal defini- 
tions of antecedent and consequent. For it is impossible that 
things should not be as a necessary proposition indicates, 
if they are as any other (proposition) indicates. Hence a 
necessary proposition is a consequent of any proposition. It 
follows therefore that this consequence is valid: 'a man runs, 
therefore God exists', or '(therefore) ass is an animal'. 

200 



CONSEQUENCES 

The proofs of these two rules are very typical of the Scholastic 
approach to propositional logic and show how different it is both to 
that of the Megarian-Stoics and that of the moderns. The essential 

scholastic point is that a consequence does not unite two states of 
affairs but two propositions (in the scholastic sense, which includes 
the mental propositions, cf. 26.03). Let 'P' be the name of the 
proposition expressing the state of affairs p, and l Q' the name of 
the proposition expressing the state of affairs q, the proof of the 
first consequence can be presented thus: 

As axiom is presupposed 

31.151 If p cannot be the case then (p and q) cannot be the 
case. 

Then the process is: 

(1) P is impossible (hypothesis) 

(2) p cannot be the case (by (1) and the definition of 

impossibility) 

(3) (p and not q) cannot be the case (by (2) and 31.151) 

(4) Q is the consequent of P (by the definition and (3)) 

(5) Q follows from P (by definition) 
And this was to be proved, Q being any proposition. 

Thus we can see that a metalogical thesis about a relationship 
(consequence) between propositions is proved through reduction to 
logical laws concerning relationships between states of affairs. 

31.16 Third rule: (1) From any proposition there follows 
every other whose contradictory opposite is incompatible 
with it (the first). And (2) from no proposition does there 
follow another whose contradictory opposite is compatible 
with it, where (the expression) 'a proposition is compatible 
with another' is to be understood in the sense that the state of 
affairs (sic esse) which the one indicates is compatible with that 
which the other indicates. . . . 

The first part of the rule is proved (thus) : Let us suppose 
that the proposition B is incompatible with the proposition A. 
I say (then) that from A there follows the contradictory 
opposite of B, i.e. not-B. This is evident, for A and B are 
incompatible and therefore (either A) is impossible, so that 
every proposition follows from it, by the first rule; or A is 
possible, then necessarily if A is the case, either B or not-B is 
the case, since one part of a pair of contradictory opposites is 
always true. But it is impossible that if A is the case, B is the 

201 



SCHOLASTIC LOGIC 

case, by hypothesis. Therefore it is necessary that if A is the 
case, not-B is the case. Therefore not-B follows from A. 

The second part of the rule is proved (thus) : if A and 
not-B are true together, then this holds: If A is the case, 
B is not the case. But since B and not-B are not true together, 
it is possible that if A is the case, B is not the case. Therefore 
B does not follow from A. 

31.17 Fourth rule: for every valid consequence, from the 
contradictory opposite of the consequent there follows the 
contradictory opposite of the antecedent. This is evident, 
since on the supposition that B follows from A, I say that 
not-^4 follows from not-B. For either it is so, or it is possible 
that A and not-B are true together, by the previous rule. But 
it is necessary that if A is the case, B is the case. Therefore B 
and not-B will be true together, which is impossible, by the 
accepted (communis) principle 'it is impossible that two contra- 
dictories should be true together'. . . . 

31.18 Fifth rule: if B follows from A, and C from B, then (1) 
C follows from A ; and (2) C follows from everything from 
which B follows; and (3) what does not follow from A, does 
not follow from B; and (4) from everything from which C does 
not follow, B too does not follow. That is to say, in current 
terms, all the (following) consequences are valid: (1) Whatever 
follows from the consequent follows from the antecedent; 
(2) The consequent of this consequence follows from all that 
from which the antecedent follows; (3) What does not follow 
from the antecedent does not follow from the consequent; 
(4) The antecedent does not follow from that from which the 
consequent does not follow. This rule has four parts. 

The first (part) is: If B follows from A, and C from B, then 
C follows from A. For on the supposition that B follows from 
A, if things are as A indicates, they are also as B indicates, by 
the nominal definition of antecedent and consequent. And 
on the supposition that C follows from B, if things are as B 
indicates, they are also as C indicates. Therefore, if things 
are as A indicates, they are also as C indicates. And accordingly 
C follows from A. 

The second part is evident, since nothing from which B 
follows can be the case if B is not the case ; and as B cannot be 
the case if C is not the case, it follows also: C follows from all 
from which B follows. And by 'being the case' is to be under- 
stood : being as B indicates .... 

202 



CONSEQUENCES 

31.19 Sixth rule: (1) It is impossible that false follows from 
true. (2) It is also impossible that from possible follows 
impossible. (3) It is also impossible that a not necessary 
proposition follows from a necessary one. (The first part) 
is evident by the nominal definition of antecedent and conse- 
quent. For if things are as the antecedent indicates, they are 
also as the consequent indicates, and accordingly, when the 
antecedent is true, the consequent is true and not false. The 
second part is evident, for if things can be as the antecedent 
indicates, they can also be as the consequent indicates; and 
accordingly, when the antecedent is possible, the consequent 
also (is possible). The third part is evident, for if things 
necessarily are as the antecedent indicates, they must also 
(necessarily) be as the consequent indicates. 

31.20 There follows from this rule: (1) if the consequent 
of a consequence is false, its antecedent is also false ; (2) further, 
if the consequent of a consequence is impossible, its antece- 
dent is also impossible ; (3) further if the consequent of a con- 
sequence is not necessary, its antecedent also is not necessary. 

And I purposely (notanter) say, 'if the consequent is not 
possible' and not 'if the consequent is not possibly true', since 
in this (consequence) : 'every proposition is affirmative, 
therefore no proposition is negative', its antecedent is possible 
and its consequent too is possible, but although it is possible, it 
is impossible that it be true, as was said above. And yet true 
can follow from false, and possible can follow from impossible, 
and necessary can follow from not necessary, as is evident 
from Aristotle in the second (book) of the Prior (Analytics, 
ch. 2) (16.32) 

31.21 Seventh rule: if B follows from A together with one 
or more additional necessary propositions, then B follows 
from A alone. Proof: B is either necessary or not necessary. If 
it is necessary, it follows from A alone, by the second rule, 
since the necessary follows from any (proposition). But if B 
is not necessary, then A is either possible or impossible. 
Suppose A is impossible then again B follows from A alone as 
also from A with an additional necessary proposition, by the 
first rule. Since from the impossible, anything follows. But 
suppose A is possible, then if A is the case it is impossible that 
B is not the case, or, if A is the case it is possible that B is 
not the case. On the first supposition, B follows from A alone, 
as also from A with an additional necessary proposition, by the 

203 



SCHOLASTIC LOGIC 

nominal definition of antecedent and consequent. But sup- 
posing that if A is the case it is possible that B is not the case, 
then if A is the case, A and the additional necessary pro- 
position must be true together. For it is impossible that* A 
should not be the case, since it is not possible that if A is the 
case, A is not the case. And accordingly, granted that A is the 
case, i.e. granted that things are as A indicates, it is necessary 
that they should be as A and the additional necessary pro- 
position indicate. Therefore from A there follow A and the 
additional necessary proposition. And as B follows from A 
and the additional necessary proposition, one obtains the 
probandum by means of the first part of this rule, (viz.) 
that B follows from A alone, which was to be proved. 

The rule could be formulated: 

31.211 If C is necessary, then : if B follows from A and C, B 
follows from A alone, 

and the proof is contained in the words: 'if A is the case, A and the 
additional necessary proposition must be the case' and the subse- 
quent justification. For in fact, if C is necessary, then if we have A, 
we have A and C, and then if B follows from A and C, B follows 
from A. The passage previous to the words just quoted is therefore 
superfluous, but it has been retained as characteristic of the scho- 
lastic approach. 

31.22 Eighth rule: every consequence of this kind is for- 
mal: 'Socrates exists, and Socrates does not exist, therefore a 
stick stands in the corner'. Proof: By formal consequence it 
follows: 'Socrates exists and Socrates does not exist, therefore 
Socrates exists', from a complete copulative proposition to 
one of its parts. Further it follows: 'Socrates exists and 
Socrates does not exist, therefore Socrates does not exist' by 
the same rule. And it further follows: 'Socrates exists, there- 
fore either Socrates exists or a stick stands in the corner'. 
The consequence holds, since from every categorical propo- 
sition a disjunctive proposition is deducible (infert) of which 
it is a part. And then again: 'Socrates exists and Socrates does 
not exist; therefore (by the second part of this copulative 
proposition): Socrates does not exist; therefore a stick stands 
in the corner'. The consequence holds since the consequence 
is formal from a disjunctive with the denial (destrudio) of one 

* Omitting necessariam. 

204 



CONSEQUENCES 

of its parts to the other. And so every proposition similar in 
form to this would be a valid consequence if it were formed. 
This rule is usually expressed in the following words : 'from 
every copulative consisting of contradictorily opposed parts, 
there follows any other (proposition by) formal consequence'. 

This text is undoubtedly one of the peaks of scholastic prepo- 
sitional logic. Both the rule and its proof were part of the scholastic 
capital. It is to be found in Pseudo-Scotus in the form: 

31.23 From every proposition evidently implying a con- 
tradiction, any other formally follows. So there follows for 
instance: 'Socrates runs and Socrates does not run, therefore 
you are at Rome.' 

The proof in 31.22 relies on the following laws as axioms, which 
are expressly formulated : 

31.221 If P and Q then P. 

31.222 If P and Q then Q. 

31.223 If P then, P or Q. 

31.224 If P or Q, then, if not-P then Q. 

The proof runs thus: 

(1) P and not-P (hypothesis) 

(2) P (by (1) and 31.221 with sub- 

stitution of 'not-P' for l Q') 

(3) P or Q (by (2) and 31.223 (cf. 31.10)) 

(4) not-P (by (1) and 31.222 with sub- 

stitution of 'not-P' for l Q') 

(5) Q (by (3), (4) and 31.224) 
And this was the probandum, Q being any proposition at all. 

Of the laws used in this proof, 31.221-2 are to be found in Ockham 
(31.24) and were also familiar to Paul of Venice. 31.223 is the modern 
law of the factor, accepted by the Scholastics from the time of Kil- 
wardby (31.10). 31.224 is the later modus tollendo ponens, ana- 
logous to the fifth indemonstrable of the Stoics (22.08), but using 
non-exclusive disjunction. 

D. PAUL OF VENICE 

Next we give some rules for copulative propositions, from Paul 
of Venice. 

31.25 For the truth of an affirmative copulative (propo- 
sition) there is required and suffices the truth of both parts 
of the copulative. . . . 

31.26 A corollary from this rule is the second: that for the 

205 



SCHOLASTIC LOGIC 

falsity of an affirmative copulative the falsity of one of its parts 
is sufficient. . . . 

31.27 The third rule is this: for the possibility of the copu- 
lative it is required and suffices that each of its principal parts 
is possible and each is compossible with each - or if there are 
more than two, with all. . . . 

31.28 From this follows the fourth, viz.: for the impos- 
sibility of a copulative it is sufficient and requisite that one of 
its principal parts be impossible or that one be not compossible 
with the other, or the others. . . . 

31.29 The fifth rule is this: for the necessity of an affir- 
mative copulative, the necessity of every one of its parts. . . . 

31.30 From this rule follows the sixth: that for the contin- 
gence of a copulative it is required and suffices that one of its 
categorical principal parts be contingent and compossible with 
the other, or with all others if there are more than two. 

Similar rules for 'known', 'known as true', 'credible' follow. 
For disjunctives, the same author gives the following rules, 
among others: 

31.31 From what has been said (cf. 31.223) there follow 
four corollaries. The first is: if there is an affirmative dis- 
junctive . . . composed of two categoricals of which one is 
superordinate to the other by reason of a term or terms in it, 
the argument is valid to the superordinate part; it follows 
e.g.: you run or you are in motion, therefore you are in 
motion. . . . 

31.32 The second corollary is this: if there is a disjunctive 
consisting of two categoricals of which one is possible, the 
other impossible, the argument to the possible part is valid. 
Hence it follows validly: 'there is no God, or you do not exist, 
therefore you do not exist'; 'you are an ass or you run, 
therefore you run'. 

31.33 The third corollary is this: if there be formed a 
disjunctive of two categoricals that are equivalent (convertibi- 
libus), the argument to each of them is valid, for it follows 
validly: 'there is no God or man is an ass, therefore man is an 
ass'. And from the same antecedent it follows that there is no 
God, since those categoricals, being impossible, are equivalent. 
It further follows: 'you are a man or you are risible, therefore 
you are risible', and it also follows that you are a man. 

31.34 The fourth corollary is this: if a disjunctive be 

206 



CONSEQUENCES 

formed of two categoricals of which one is necessary and the 
other contingent, the argument is valid to the necessary part. 
Hence it follows validly: 'you run or God exists, therefore 
God exists'. And it is not strange that all such consequences 
hold, for the consequent follows of itself immediately (con- 
tinue) from each part of the disjunctive, hence it must follow 
too from the disjunctives themselves. 

31.35 The eighth principal rule is this: from an affirmative 
disjunctive ... to the negative copulative formed of the 
contradictories of the parts of the disjunctive is a valid 
argument. The proof is that the affirmative copulative formed 
of the contradictories of the parts of the disjunctive contradicts 
the disjunctive, therefore the contradictory of that copulative, 
formed by prefixing a negative, follows from the disjunctive. 
For example 'you run or you are in motion, therefore: not, 
you do not run and you are not in motion' ; 'God exists or no 
man is an ass, therefore: not, there is no God and man is an 
ass'. Those consequences are evident, since the opposites 
of the consequents are incompatible with their antecedents, 
as has been said. 

31.36 From that rule there follows as a corollary that from 
an affirmative copulative ... to a negative disjunctive 
formed of the contradictories of the parts of the copulative is 
a valid argument. Hence it follows validly: 'you are a man 
and you are an animal, therefore: not, you are not a man or 
you are not an animal'. Similarly it follows: 'you are not a 
goat and you are not an ass, therefore : not*, you are a goat or 
you are an ass'. . . . 

The last two texts contain two of the so-called 'de Morgan ' laws. 
So far as is known, they first occur in Ockham (31.37) and Burleigh 
(31.38)**. The latter gives them in the form of equivalences. 

31.39 If one argues from an affirmative conditional, 
characterized (denominaia) by 'if, to a disjunctive consisting 
of the contradictory of the antecedent and the consequent* 
of the conditional the consequence is formal. The proof is 
that this consequence is formal : 'if you are a man you are an 
animal; therefore, you are not a man or you are an animal'. 
And no one example is more cogent than another, therefore all 
are valid consequences. 

* Adding non. 
* * But Peter of Spain ( Traclalus Syncategorematum) has the doctrine of 31.36. 
* * * Reading consequente for consequentis. 

207 



SCHOLASTIC LOGIC 
E. RULES OF CONSEQUENCES UT NUNC 

Buridan writes: 

31.40 And it is to be noted that a proportionate conclusion 
is to be posited concerning consequences ut nunc (i.e. propor- 
tionate to that concerning simple consequences), viz. that 
from every false proposition there follows every other by a 
consequence ut nunc, because it is impossible that things being 
as they now are a proposition which is true should not be true. 
And so it is not impossible that it should be true, however 
anything else may not be true. And when the talk is of the 
past or future, then it can be called a consequence ut nunc, or 
however else you like to call it, e.g. it follows by a consequence 
ut nunc or ut tunc or even nunc per tunc: 'if Antichrist will not 
be generated, Aristotle never existed'. For though it be simply 
true that it is possible that Antichrist will not exist, yet it is 
impossible that he will not exist when things are going to be 
as they will be ; for he will exist, and it is impossible that he 
will exist and that he will not exist. 

We have here first of all the two classical 'paradoxical' laws of 
material implication: 

31.401 If P is false then Q follows from P. 

31.402 If P is true, then P follows from Q. 

Buridan provides an example of (substitution in) the first of 
these laws, putting the proposition 'Antichrist will not be generated' 
for l P\ and 'Aristotle never existed' for l Q\ The first proposition 
is, absolutely speaking, possible, so this cannot be a case of simple 
consequence (cf. the first rule, 31.14 supra), for that would require 
it to be absolutely impossible. But the consequence holds if taken 
ut nunc, since the proposition 'Antichrist will not be generated' will 
in fact be impossible in what will be the circumstances. Hence 
we have impossibility for that time (ut tunc) and so a consequence 
that holds for that time. 

This shows that even consequence for a given time is defined by 
means of impossibility. The difference between it and simple conse- 
quence consists only in the kind of impossibility, absolute (for all 
times and circumstances) in the case of simple consequence, con- 
ditioned in that of consequence for a given time. 

But impossibility ut nunc is defined as simple non-existence, and 
so the proposition 'Antichrist will not be generated' can be reckoned 
as impossible since Antichrist will in fact be generated. It follows 
that consequence ut nunc can be defined without the help of the 
modal functor, a proposition ut nunc being impossible simply when 
it is false. 

208 



CONSEQUENCES 

Another law of consequence ut nunc comes from the same text of 
Buridan : 

31.41 If a conclusion follows from a proposition together 
with one or more additional propositions, the same conclusion 
follows from that proposition alone by a consequence ut nunc. 

This rule is analogous to the seventh rule given above for simple 
consequence (31.21), which shovvs that the whole system of simple 
consequences can be transformed into a system of consequences 
ut nunc by everywhere replacing 'necessary' by 'true' and 'impossi- 
ble' by 'false' and similar simplifications. 

Finally we remark that Buridan, so far as is now known, is the 
only scholastic logician to develop laws of consequence ut nunc, 
though he devoted much less space to them than to simple conse- 
quences. In Paul of Venice the subject of consequence ut nunc seems 
to have dropped out completely. 



209 



III. LOGIC OF TERMS 



§32. ASSERTORIC SYLLOGISTIC 

Contrary to a widespread opinion, the assertoric syllogistic was 
not only not the only, it was not even the chief subject of scholastic 
logic. The Scholastics, like most of the Commentators (24.271 ff.), 
thought of syllogisms as rules (cf. 30.11) rather than conditional 
propositions. The domain of syllogisms received a significant exten- 
sion through the introduction of singular terms already in Ockham's 
time. But the new formulae thus derived will here be given separately 
under the heading 'Other Formulae' (§ 34) since they effect an 
essential alteration in the Aristotelian syllogistic. In the present 
section we shall confine ourselves to that part of the scholastic 
treatment which can still be deemed Aristotelian. Here, too, 
everything is treated purely metalogically (except in some early 
logicians such as Albert the Great), but that is quite in the Aristo- 
telian tradition (14.25ff.). 

The most important contributions are these: (1) the devising of 
numerous mnemonics for the syllogistic moods and their inter- 
relationships, culminating in the pons asinorum. (2) The systematic 
introduction and thorough investigation of the fourth figure. (3) 
The position and investigation of the problem of the null class, which 
has already received mention in connection with appellation (§ 28, B). 

A. EARLY MNEMONICS 

L. Minio-Paluello has recently made the big discovery of an early 
attempt to construct syllogistic mnemonics, in a MS of the early 
13th century. This deprives of its last claims to credibility Prantl's 
legend of the Byzantine origin of such mnemonics. * The essentials 
are these: 

32.06 It is to be remarked that there are certain notations 
(notulae) for signifying the moods. . . . The four letters e, i, o, u 
signify universal affirmatives, and the four letters /, m, n, r 
signify universal negatives, and the three** a, s, t signify 



* Carl Prantl, relying on a single MS, ascribed to Michael Psellus (1018 to 
1078/96) the 'Lvvotyit; elq ttjv 'AptaTOTeXou<; Xoyix^v eTuarrjpgv in which such 
mnemonics occur, and stated that the Summulae of Peter of Spain was a trans- 
lation from that (32.01). In that opinion he was the victim of a great mistake, 
since M. Grabmann (32.02) following C. Thurot (32.03), V. Rose (32.04) and 
R. Stapper (32.05) has shown the Hvvotyic, to be by George Scholarios (1400-1464) 
and a translation of the Summulae. 
* * Reading '3' for '4'. 

210 



ASSERTORIC SYLLOGISTIC 

particular affirmatives, and b, c, d signify particular negatives. 
So the moods of the first figure are shown in the following 
verse: uio, non, est (tost), lac, uia, mel, uas, erp, arc. Thus the 
first mood of the first figure, signified by the notation uio, 
consists of a first universal affirmative and a subsequent 
universal affirmative, (and) concludes to a universal affirma- 
tive; e.g.: All man is animal and all risible is man, therefore 
all risible is animal. . . . The moods of the second figure are 
shown in the following verse: ren, erm, vac* , obd. . . . The 
moods of the third figure are shown in the following verse: 
eua, nee, aut, esa, due, nac. 

This is a very primitive technique, but at least it shows that the 
highly developed terminology of Peter of Spain*** had antecedents 
in Scholasticism itself. 

We cite the relevant texts from the Summulae Logicales. 

B. BARBARA-CELARENT 

32.07 After giving a threefold division of propositions it is 
to be known that there is a threefold enquiry to be made about 
them, viz. What?, Of what kind?, How much? 'What?' 
enquires about the nature (substantia) of the proposition, 
so that to the question 'What?' is to be answered 'categorical' 
or 'hypothetical'; to 'Of what kind?' - 'affirmative' or 
'negative'; to 'How much?' - 'universal', 'particular', 
'indefinite', 'singular'. Whence the verse: 

Quae ca vel hip, qualis ne vel aff, un quanta par in sin. 
the questions being in Latin: quae?, qualis?, quanta?, and 
the answers: categorica, hypoihetica, affirmaiiva, universalis, 
particularis, infinita, singularis. 

So far as we know this is the first text in which the notions of 
quality and quantity occur. The full set of technical terms connected 
therewith, together with some others, appears in the next passage 
which resumes the doctrine of conversion. 

32.08 The conversion of propositions having both terms 
in common but with the order reversed, is threefold; viz. 

* MS has 'rachc\ 
* * Personal thanks are due to Prof. L. Minio-Paluello for telling us of this MS 
and helping to restore the text. 

* * * This can not have originated with him. Prof L. Minio-Paluello informed 
me on 24. 6. 55 that he had found the word 'Fesiino' in a MS dating at the latest 
from 1200. 

211 



SCHOLASTIC LOGIC 

simple, accidental, and by contraposition. Simple conversion is 
when the predicate is made from the subject and conversely, 
the quality and quantity remaining the same. And in this 
way are converted the universal negative and the particular 
affirmative. . . . The universal affirmative is similarly con- 
verted when the terms are equivalent (convertibilibus). . . . 

Accidental conversion is when the predicate is made from 
the subject and conversely, the quality remaining the same, 
but the quantity being changed ; and in this way the universal 
negative is converted into the particular negative, and the 
universal affirmative into the particular affirmative. . . . 

The law of accidental conversion of the universal negative is not 
in Aristotle. 

32.09 Conversion by contraposition is to make the predi- 
cate from the subject and conversely, quality and quantity 
remaining the same, but finite terms being changed to 
infinite ones. And in this way the universal affirmative is 
converted into itself and the particular negative into itself, 
e.g. 'all man is animal' - 'all non-animal is non-man'; 'some 
man is not stone' - 'some non-stone is not non-man'.* 

Hence the verses: 

A Affirms, E rEvokes**, both universal, 

/ affirms, revOkes**, both in particular. 

Simply converts fEel, accidentally EvA, 

AstO by contra(position) ; and these are all the conversions. 

The classical expressions Barbara, Celarent, etc. seem to have 
been fairly generally known about 1250. After describing the 
assertoric moods Peter of Spain introduced them thus: 

32.10 Hence the verses: 

Figure the first to every kind*** concludes, 
The second only yields negations, 
Particulars only from third figure moods. 
Barbara, celarent, darii, ferion, baralipton, 
Celantes, dabitis, fapesmo, frisesomorum. 
Cesare. camestres, festino, baroco, darapti. 
Felapto, disamis, datisi, bocardo, ferison. 

* Reading non homo. 
* * negat. 
* * * viz. problematis. 

212 



ASSERTORIC SYLLOGISTIC 

32.11 In those four verses there are twenty-one expressions 
(didiones) which so correspond to the twenty-one moods of the 
three figures that by the first expression is to be understood the 
first mood, and by the second the second, and so with the others. 
Hence the first two verses correspond to the moods of the 
first figure, but the third to the moods of the second save for 
its last expression. It is to be known therefore that by these 
four vowels, viz. A, E, 1,0 set in the aforesaid verses there 
are understood the four kinds of proposition. By the vowel A 
is understood the universal affirmative, by E the universal 
negative, by / the particular affirmative, by the particular 
negative. 

32.12 Further it is to be known that in each expression 
there are three syllables representing three propositions, and 
if there is anything extra it is superfluous, excepting M as 
will appear later. And by the first syllable is understood the 
major proposition, similarly by the second the second proposi- 
tion, and by the third the conclusion; e.g. the first expression, 
viz. Barbara, has three syllables, in each of which A is set, 
and A set three times signifies that the first mood of the first 
figure consists of two universal affirmatives concluding to a 
universal affirmative; and thus it is to be understood about 
the other expressions according to the vowels there set. 

32.13 Further it is to be known that the first four expres- 
sions of the first verse begin with these consonants, B, C, D, 
F, and all the subsequent expressions begin with the same, 
and by this is to be understood that all the subsequent moods 
beginning with B are reduced to the first mood of the first 
figure, with C to the second, with D to the third, with F to 
the fourth. 

32.14 Further it is to be known that wherever S is put in 
these expressions, it signifies that the proposition understood 
by means of the vowel immediately preceding should be 
converted simply. And by P is signified that the proposition 
which is understood by means of the vowel immediately 
preceding is to be converted accidentally. And wherever M 
is put, it signifies that transposition of the premisses is to be 
effected. Transposition is to make the major minor and 
conversely. And where C is put, it signifies that the mood 
understood by means of that expression is to be reduced 
per impossibile. 

Whence the verses: 

213 



SCHOLASTIC LOGIC 

S enjoins simple conversion, per accidens P, 
Transpose with M, ad impossibile C. 

George Scholarios's Greek version of the four last verses of 32.10 
is not without interest (cf. 32.02 ff.): 

32.15 Tpa^fjiaTa Kypa^e ypacpiSi re^vixoc;, (I) 
rpa[X(JLacr!,v fe'ra^e x^? lGl rcapOevos ispov (la) 
"Eypoc^s y.oi'zzjz uiTpiov a^oXov. (II) 
"Ktzolgi aOevapcx; taaxt? a<j7ci§i 6(xaXo<; (pepwjTos. (Ill) 

Unlike the Latin ones, these verses are meaningful, and can be 
rendered: 

Letters there wrote with a style a scholar, 

With letters there composed for the Graces a maiden a dedica- 
tion. 

She wrote: Cleave to the moderate, un-wrathful (man). 

In all, that strength which like a shield is well-proportioned is 

the best. 

The names Barbara, Celareni etc. have survived the era of Scho- 
lasticism and are still in use today, unlike many other syllogistic 
mnemonics. We give some examples of these others, and first some 
which concern the technique of reducing moods of the second and 
third figures (also of the 'indirect' moods of the first figure : 17.111 ff.) 
to moods of the first. For that purpose, Jodoc Trutfeder at the 
beginning of the 16th century gave these expressions: 

32.16 Baralipton Nes- Celareni 



Celantes 


ci- 


Darii 


Dab His 


e- 


Celareni 


Fapesmo 


ba- 


Barbara 


Frisemom 


tis. 


Darii 


Cesare 


0- 


Ferio 


Camestres 


di- 


Darii 


Festino 


e- 


Celareni 


Baroco 


bam. 


Barbara 


Darapli 


Le- 


Celareni 


Felapton 


va- 


Barbara 


Disamis 


re. 


Celareni 


Datisi 


Bo- 


Ferio 


Bocardo 


man- 


Barbara 


Ferison 


nis. 


Darii 



These expressions serve for the indirect process of Aristotle 
(§ 14, D). Thus for instance from Celareni, by putting the contra- 
dictory opposite (/) of its conclusion (E) for the minor premiss (A) 
there is concluded the contradictory of the latter (0), and one has 

214 



ASSERTORIC SYLLOGISTIC 

Festino. Applying this treatment to the major premiss one has 
Disamis. Evidently some further processes must be employed to 
get a few further moods. E.g. to get Felapton and Darapti from 
Barbara and Celarent respectively, one must first deduce Barbari 
and Celaronl (24.271 f.). So too in the case of Dabilis (24.273). 

Further mnemonics that were used will he mentioned later 
(32.24 and 32.38). 

C. BARBARI-CELARONT 

Similar mnemonic expressions are found for the so-called 'sub- 
alternate' moods (24.271-24.281). A complete list with names 
appears in a text of Peter of Mantua that is in other respects very 
defective: 

32.17 . . . the first (formula) ... is usually signified by the 
expression Barbara. The second formula has premisses arranged 
in the way described which conclude to the particular affir- 
mative or indefinite of the consequent of the first formula 
that we posited, and this we are wont to call Barbari. . . . 

From the aforesaid (premisses) can also be concluded the 
particular negative of (i.e. corresponding to) the consequent 
of the aforesaid formula, which (new) formula we can call 
Celaronl. . . . 

The eighth formula, which is called Baralipton, follows 
from Barbari, by conversion of its conclusion. . . . 

The ninth formula is called Celantes . . . from it follows the 
tenth, which is called Celantos; it concludes to a particular 
or indefinite conclusion. . . . 

The second mood (of the second figure) can be gained from 
the aforesaid premisses (of the mood Cesare) by concluding 
to the particular that corresponds to (Cesare's) conclusion, 
and is signified by the expression Cesaro. . . . 

The next formula ... is usually called Cameslres. From it 
there follows another formula which we call Camestro. 

So Peter of Mantua has five subalternate moods besides the 
nineteen moods of Peter of Spain, in fact the full twenty-four. But 
he has many others as well, commonly forming an 'indirect' mood 
corresponding to each of the others (applying the Aristotelian rules 
of § 13), e.g. a Cesares corresponding to Cesare. Cesares would look 
like this: 

No man is stone ; 
All marble is stone ; 
Therefore no man is marble. 

215 



SCHOLASTIC LOGIC 

Contrast the following in Camestres : 
All marble is stone; 
No man is stone; 
Therefore no man is marble. 

The only difference between the two is in the order of the premisses, 
and to reckon them as distinct moods is an extreme of formalism. 
Peter of Mantua further forms such moods as Barocos, with the 
O-conclusion of Baroco converted (!) and other false formulas.* 

D. THE FOURTH FIGURE 
1. Among the Latins 

We know of no scholastic logical text in Latin where the fourth 
figure in the modern sense can be found, though all logicians of the 
period develop the 'indirect moods of the first'. They are mostly 
aware of a fourth figure, but treat it as not distinct from the first, e.g. 
Albert the Great (32.18), Shyreswood (32.19), Ockham (32.20), 
Pseudo-Scotus (32.21), Albert of Saxony (32.22), Paul of Venice 
(32.23). We give an instance from Albert of Saxony. 

32.22 (The syllogism is constituted) in a fourth way if 
the middle is predicated in the first premiss, subjected in the 
second. . . . But it is to be noted that the first figure differs 
from the fourth only by interchange of premisses which does 
nothing towards the deducibility or non-deducibility of the 
conclusion. 

Some later logicians do recognize a 'fourth figure' but this again 
is not the modern one; only the first with interchanged premisses, as 
in the last text. This is very clear in Peter Tarteret and Peter of 
Mantua. We quote the first: 

32.24 First (dictum): Taking 'figure' in a wide sense, the 
fourth figure is no different from the first but contained under 
it. Second: taking 'first figure' in a specific sense, a fourth 
figure is to be posited distinct from the first; and the fourth 
figure consists in this, that the middle is predicate in the 
major premiss, subject in the minor, e.g. 'all man is animal; 
all animal is substance; therefore all man is substance'. 

* It should be understood in respect of this and the following sub-section 
that after Peter of Spain (generalizing the method of Boethius and Shyreswood 
for the second and third figures) it was usual to define the major and minor 
premisses as the first and second stated, and the extreme terms with reference 
to the premisses, not the conclusion. 48 moods in 4 figures can be (and some- 
times were) correctly distinguished on this basis. 'Classical' failure to distinguish 
this from the method of Albalag (which goes back to Philoponus) resulted in 
many inconsistencies. (Ed.) 

216 



ASSERTORIG SYLLOGISTIC 

Third : there are four moods of the fourth figure, viz. Bamana, 
Camene, Dimari, and Fimeno. They are reduced to the first 
figure by mere exchange of premisses. 
Peter of Mantua also has Bamana etc. 

2. In Albalag 

Yet a clearly formulated doctrine of the 'genuine' fourth figure is 
to be found in a 13th century text- of the Jewish philosopher Alba- 
lag.* This text, like the foregoing, seems to have been without 
influence on the development of logic in the Middle Ages. It was 
recently discovered by Dr. G. Vajda and has never been translated 
into Latin. We quote it here at length for its originality, and because 
it is instructive about the level of logic at that time. 

32.25 In my opinion there must be four figures. For the 
middle term can be subject in one of the two premisses and 
predicate in the other in two ways: (1) the middle term is 
subject in the minor, predicate in the major, (2) it is predicate 
in the minor, subject in the major. The ancients only con- 
sidered the second arrangement and called it the 'first figure'. 
This admits of four moods which can yield a conclusion. But 
the first arrangement, which I have found, admits of five 
moods which can yield a conclusion. . . . 

32.26 We say then that this new figure is subject to three 
conditions: (1) one of its premisses must be affirmative, the 
other universal; (2) if the minor premiss is affirmative, the 
major will be universal; (3) if the major is particular, the 
minor will be affirmative. 

The conditions exclude eleven of the sixteen (theoretically 
possible) moods; there remain therefore five which can yield 
a conclusion. 

32.27 (1) The minor particular affirmative, the major 
universal negative : 

Some white is animal. 

No raven is white. 

Some animal is not raven. 
Then one can convert the minor particular affirmative and 
the major universal negative and say: 

Some animal is white, 

No white is raven, 
which yields the third mood of the first figure. 

* This was kindly put at my disposal, together with a French translation, by 
Dr. G. Vajda. 

217 



SCHOLASTIC LOGIC 

32.28 (2) The minor universal affirmative, the major 
universal negative: 

All man is animal. 

No man is raven. 

Some animal is not raven. 
Then one can get back to the third mood of the first figure 
by converting both premisses. 

32.29 (3) The minor universal negative, the major universal 
affirmative : 

No man is stone. 

Every speaker is man. 

No stone is speaker. 
Exchanging the minor and major with one another, one comes 
back to the second mood of the first figure, of which the 
conclusion will be: 'No speaker is stone', and one only needs 
to convert this to obtain 'No stone is speaker'. 

32.30 (4) Two affirmatives: 
All composite is not eternal. 
All body is composite. 
Some not eternal is body. 

Here one can interchange the minor and major premisses 
and reach the first figure, with conclusion: 'All body is not 
eternal' which can be converted to 'Some not eternal is body'. 

32.31 (5) The minor universal affirmative, the major 
particular affirmative : 

All man is speaker. 

Some white is man. 

Some speaker is white. 
If one interchanges the minor and major premisses, one 
concludes in the first figure: 'some white is speaker' which 
will be converted as above. . . . 

32.32 . . . the syllogism is formed with reference to a 
determinate proposition which is first established and laid 
down in the mind, and the truth of which one then tries to 
justify and manifest by means of the syllogism. Of the 
premisses, that containing the term which is predicate of 
this proposition is the major, that containing the subject 
is the minor. 

Albalag here presents the modern definition of the syllogistic terms, 
not according to their extension, but formally, according to their pla- 
ces in the conclusion. The modern names of the moods he introduces 
are: Fresison, Fesapo, Calemes, Bamalip and Dimaris (cf. § 36, F). 

218 



assertory syi.i.oojsth; 



There is missing only that corresponding to Peter of Mantua's 
Celantos (32.17), viz. Calemop. Albalag also formulates the general 

rules of the fourth figure, and uses the combinatorial method. 



E. COMBINATORIAL METHOD 

In Albert the Great we find a procedure taken over from the 
Arabs (32.33) by which all possible moods of the syllogism are first 
determined combinatorially, and the invalid ones then discarded. 
The relevant text runs: 

32.34 It is to be known that with such an ordering of 
terms and arrangement of premisses (propositionum) sixteen 
conjugations result, yielded by the quantity and quality 
of the premisses. For if the middle is subject in the major and 
predicate in the minor, either (1) both premisses are universal, 
or (2) both are particular, or one (is) universal and the other 
particular and this in two ways: for either (3) the major is 
universal and the minor particular, or (4) conversely the 
major particular and the minor universal; these are the four 
gained by combinations of quantity. When each is multiplied 
by four in respect of affirmation and denial, there are sixteen 
conjugations in all, thus: if both (premisses) are universal 
either (1) both are affirmative, or (2) both negative, or (3) 
the major is affirmative and the minor negative, or con- 
versely (4) the major is negative and the minor affirmative: 
and there are four conjugations. But if both are particular, 
there are again four conjugations . . . etc. 

We may compare with that the text of Albalag (32.25 ff.). Kil- 
wardby uses similar methods. 



F. INVENTIO MEDII, PONS ASINORUM 

The Aristotelian doctrine of the inveniio medii (14.29) was keenly 
studied by the Scholastics, and the schema of Philoponus (24.35) was 
not only taken over, but also further developed. It is to be found 
as early as Albert the Great, who probably found it in Averroes 
(32.35) ; Albert's version differs from that of Philoponus and Aver- 
roes only in the particular formulae employed. But as we find it in 
him, it became the foundation of the famous pons asinorum. so 
that it must be given in this form as well : 



219 



SCHOLASTIC LOGIC 



32.36 

All Pis M.) AB 



(NoPisM.) AD 



(All Mis P.) AC 




FE (All S is M. 



HE (NoSisM.) 



(All MisS.) 



The further development of this figure is the pons asinorum, 
which must have been known to George of Brussels (32.37) since 
Thomas Bricot in a commentary on George's lectures gives the 
mnemonic words for it with the following explanation: 

32.38 When the letters A, E, I, are put in the third 
syllable they signify the quality and quantity of the con- 
clusion to be drawn. . . . When the letters A and E are put 
in the first or second syllable, A signifies the predicate and E 
the subject. And each of the letters can be accompanied by 
three consonants; A with B, C, D, and then B signifies that 
the middle should follow on the predicate, C that it should 
be antecedent, D that it should be extraneous. Similarly E 
is accompanied by F, G, H, and then F signifies that the 
middle should follow on the subject, G that it should be 
antecedent, H that it should be extraneous. As is made 
clear in these verses: 

jB's the subject, F its sequent, G precedent, F outside; 

^4's the predicate, B its sequent, C precedent, F outside. 
. . . To conclude to a universal affirmative, a middle is to 
be taken which is sequent to the subject and antecedent to 
the predicate ; and this is shown by Fecana. ... To conclude 
a particular affirmative in Darapti, Disamis and Dalisi, a 
middle is to be taken which is antecedent to both extremes, 
as is made clear by Cageti. ... To conclude to a universal 
negative in Celarent or Cesare a middle is to be taken which 



220 



frunpergUiitaunipti y ■ iituuti 

Pontrm u-.*Jrjura (v'rrtrrtuiido cidci't 
Impcdichicoo.l •* fenfm firm tc Jjc oc a!tog 
In doctis ulcus cl't ilhi nulla lalus 



Horrrf rqaus talcm ui ;J.n i nc »itUl?«aC1 
Dun gradirurccrnrri' ftc 'kc ire potent 
Nod iguur rurfum dira ueuant ^finorum 
(^.ui (cdcos retro nunc rcminrrcuolo. 



■fhedtutum 




/Mmus . mratjboin.dump^ns.opanerrrmu Aims, Hcti mrqd feciSruo nrcfbftmihiqie^ 
Kficns 5»tiiUiKuad A lab«,uratoic g tcdia* cadca Aux.l.uin.mon.corqua.dcdttWFOiOf. 



Pons Asinorum after Peter Tarteret (32.39 



ASSERTORIC SYLLOGISTIC 

is extraneous to the predicate and consequent on the subject, 
as is made clear by Dafenes. But if the inference is to be in 
Cameslres, the middle must be extraneous to the subject and 
consequent on the predicate, as is made clear by Hebare. . . . 
For concluding to a particular negative in the third figure, 
the middle should be antecedent to the subject and extra- 
neous to the predicate, as is made clear by Gedaco. ... To 
conclude indirectly to a particular affirmative, the middle 
must be antecedent to the subject and consequent on the 
predicate, as is made clear by Gebali. 

The mnemonic expressions given here employ the letters of 
Albert's figure, so that each expression corresponds to a line of the 
figure having the syllables of the mnemonic at its ends, a further 
syllable being added to show the quantity and quality of the 
corresponding conclusion. So this text of George's both illustrates 
the pons asinorum and helps to explain Albert's figure. 

The pons asinorum itself we have found only in Peter Tarteret, 
with the following introduction: 

32.39 That the art of finding the middle may be easy, 
clear and evident to all, the following figure is composed 
(ponitur) to explain it. Because of its apparent difficulty it 
is commonly called the 'asses' bridge' (pons asinorum) though 
it can become familiar and clear to all if what is said in this 
section (passu) is understood. 

We give the schema itself in the preceding illustration. 

G. THE PROBLEM OF THE NULL GLASS 
The problem of the null class, i.e. of the laws of subalternation, 
accidental conversion (14.12, 32.08) and the syllogistic moods 
dependent on them, has been much discussed in recent times 
(46.01 ff.). It was already posed in the fourteenth century and solved 
by means of the doctrines of supposition and appellation. We give 
three series of texts, the first attributed to St. Vincent Ferrer, the 
second from Paul of Venice, the third from a neo-scholastic of the 
17th century, a contemporary of Descartes, John of St. Thomas. 
Each gives a different solution. 

1. St. Vincent Ferrer 

32.40 Under every subject having natural supposition 
copulative descent can be made, with respect to the predicate, 
to all its supposita, whether such a subject supposes discretely* 

* Reading discrete for difinite. 

221 



SCHOLASTIC LOGIC 

or particularly or universally. Therefore it follows validly: 
Man is risible, therefore this man is risible, and that 
man. . . . 

32.41 But against this rule there are many objections. . . . 
(Sixth objection:). ... In the propositions 'rain is water 
falling in drops', 'thunder is a noise in the clouds', the subjects 
have natural supposition. Yet it is not always permissible to 
descend in respect to the predicate to the supposita of the 
subject; for it does not follow: 'rain is water falling in drops, 
therefore this rain, and that rain, etc.'; since the antecedent 
is true even when there is no rain, as will be shortly said, and 
yet the consequent is not true nor even very intelligible, 
since when there is no rain (nulla pluvia exislenle) one cannot 
say 'this rain' or 'that rain', since a contradiction would be 
at once implied. In the same way must be judged the proposi- 
tion 'thunder is a noise etc' 

32.42 To the sixth objection it is to be said that that rule . . . 
is understood (to hold) when such a subject has supposita 
actually (in aclu) and not otherwise. For no descent can be 
made to the supposita * of anything except when it has them 
actually, since, as the objection rightly says, an evident con- 
tradiction would be implied. . . . Hence the consequence which 
concludes from a universal proposition to singular ones 
contained under it, e.g. 'every man runs, therefore Socrates 
runs and Plato runs', and 'every man is an animal, therefore 
Plato is an animal etc.', is called by some logicians 'conse- 
quence ul nunc 1 . And rightly so, since no such consequence is 
valid except for a determinate time, i.e. when Socrates and 
Plato and the other supposita actually exist. 

32.43 Against the seventh objection it is to be briefly 
said that the subject of the proposition 'the rose is sweet- 
smelling' - or as one can also put it 'the rose smells sweet' - has 
personal supposition, and it follows validly 'therefore the rose 
exists (est)'. But if one says 'the rose is odoriferous' so that 
'odoriferous' (hoc quod dicitur odorifera) expresses aptitude, 
then the subject has natural supposition and it does not 
follow: 'therefore the rose exists'. Hence being odoriferous is 
to the rose as living to mankind, and what has been said about 
the proposition 'man is living' must also be understood about 
this: 'the rose is odoriferous'. 

* Reading supposita for subjecla. 

222 



ASSERTORIC SYLLOGISTIC 

The solution of the first text (32.40-42) consists then precisely in 
the exclusion of the null class (cf. § 40, Bj, 'null' being taken as 
'actually null'. In other words: in the syllogistic every term must 
have appellation in the sense of Peter of Spain (28.13 f.). In the 

second part (32.43) it is stipulated for subalternation that the subject 
must have personal (27.17 f.) and not natural (27.14) supposition. 
This evidently presupposes that a term with personal supposition 
stands for really existent things. Thus we have the same solution 
as before, 

2. Paul of Venice 

32.44 The third rule is this: universal affirmative and 
particular or indefinite (affirmative, as also universal negative 
and particular or indefinite) negative (propositions) which 
have similarly and correctly supposing terms, are subalternate, 
and conversely, explicitly or implicitly, in the logical square 
(figura). Hence the following are subalternate: 'every man is 
an animal' and 'a certain man is an animal', and similarly: 
'no man is an animal' and 'a certain man is not an animal'. 
I say 'correctly supposing terms', since the extremes must 
explicitly or implicitly stand for just the same thing, if it is a 
case of only one suppositum, for the same things, if it is a 
case of several. And so I say that (the following) are not 
subalternate: 'every man is an animal', 'a certain man is an 
animal', since under the supposition that there are no men, 
the universal would be true, but the particular false, contrary 
to the nature of subalterns. The reason why these are not 
subalternate is that the subjects do not stand for exactly 
the same thing. The subaltern of the former is therefore : 'man 
is an animal', and if one required a particular it must be this: 
'a certain being which is a man is an animal'. 

So Paul of Venice confines himself to stating the general rule 
that both propositions in a subalternation must have subjects with 
exactly the same supposition. 

3. John of St. Thomas 

John of St. Thomas deals with the problem of conversion. 

32.45 Against the conversion of the universal affirmative, 
it is objected: the consequence 'every white man is a man, 
therefore a certain man is a white man' does not hold. For 
the antecedent is necessary, but the consequent can be false, 
in case no man in the world was white. . . . 

223 



SCHOLASTIC LOGIC 

The answer is that this (proposition) is not true in the sense 
in which the first proposition of which it is the converse is 
true. For when it is said: 'every white man is a man', with 
'is' taken accidentally, for an existing man, this proposition 
in the argument given as an example is false, and its converse 
too. But when the 'is' abstracts from time and renders the 
proposition necessary, then 'white' will not be verified in 
the sense of existence, but according to possibility, i.e. 
independently of time, in the following sense: 'every possibly 
white man is a man', presupposing that no such exists. 
Accordingly the converse must be: 'therefore a certain man 
is a man who is possibly white', and thus this is true. 

The following may serve as explanation: take the proposition (1) 
'every Swiss king is a man'. By the rules of conversion (14.11, 
32.08) we may infer (2) 'a certain man is a Swiss king'. But (1) is 
true, (2) false. Therefore the rule of conversion employed is not 
valid. To this the Scholastics would answer that in (1) 'man' evi- 
dently stands for a possible man, not for a real one; it has therefore 
no appellation in the sense of 28.13. And so if (1) is converted into 
(2), 'man' in (2) also supposes for possible men and in this sense (2) 
is true as well as (1). 

A further interesting point is that singular terms always have 
appellation (28.13), so that the Scholastics attribute to proper 
names the same property with which the moderns endow descrip- 
tions (cf. § 46). 



§33. MODAL SYLLOGISTIC 

The history of scholastic modal syllogistic has been investigated 
from the modern point of view up to and inclusive of Pseudo-Scotus 
(33.01). We know that there was more than one system of modal 
logic in the Middle Ages and can to some extent follow the devel- 
opment. 

A. ALBERT THE GREAT 

The work of Albert the Great constitutes the starting point and, 
as his own text suggests (33.02), would seem to have drawn on 
Arabic sources. To begin with, he shows much the same doctrine 
as has been ascribed above (29.09) ; cf. also 23.10, 29.12) to Thomas 
Aquinas and which is basic for the whole of Scholasticism (33.03), 
viz. the distinction of the composite (composita, de dido) and the 
divided (divisa, de re) modal proposition, i. e. between that in which 
the modal functor governs the whole dictum and that in which it 

224 



MODAL SYLLOGISTIC 

governs only a part. Later Albert gives a clear statement of the 
Aristotelian distinction of the two structure- of the modal proposi- 
tion in the divided sense: 

33.04 That the predicate A possibly belongs to the subject 
B means one of these: (1) that A possibly belongs to that 
which is B and of which B is predicated in the sense of actual 
inherence, or (2) that A possibly belongs to that to which B 
possibly belongs. 

There is added a third structure, unknown to Aristotle: 

33.05 And if someone asks why the third meaning (acceptio) 
of the contingent is not given here, viz. that whatever is 
necessarily B is possibly A, since this is used in the mixing of 
the contingent and the necessary, it is to be answered that it 
is left sufficiently clear from what else has been determined 
about the mixing of the assertoric and the contingent. 

The structure in question is this: 

For all x: if x is necessarily B, x is possibly A. 

It is significant that Albert the Great puts this doctrine at the 
beginning of the presentation of his theory of modal syllogisms in a 
special chapter entitled De dici de omni el dici de nullo in propo- 
silionibus de contingenti (33.06). What for Aristotle are marginal 
thoughts about the structure of modals by comparison with his 
main ideas (15.13), have here become fundamental. 

We find then in Albert the Great a systematization of the Ari- 
stotelian teaching about the kinds of modal functors (33.07), and a 
thorough presentation of the syllogistic of the Prior Analytics. 



B. PSEUDO-SCOTUS 

Besides the four classical modal functors Pseudo-Scotus intro- 
duces others: 'of itself (per se), 'true', 'false', 'doubtful' (dubium), 
'known' {scitum), 'opined' (opinatum), 'apparent', 'known' (notum), 
'willed' (volitum), 'preferred' (dilectum) (33.08), and so a number of 
'subjective' functors. He formulates a long series of laws of modal 
propositional logic (modal consequences), among which are the 
following: 

33.09 If the antecedent is necessary the consequent is 
necessary . . . and similarly with the other (positive) modes. 

33.10 Modal (de modo) propositions in the composite sense 
with the (negative) modes 'impossible', 'false', 'doubtful' are 
not convertible like assertoric ones. Proof: for otherwise the 

225 



SCHOLASTIC LOGIC 

(following) rules would be true : 'if the antecedent is impossible, 
the consequent is impossible', 'if the antecedent is doubtful, 
the consequent is doubtful' ; but they are false. . . . 

33.11 It follows: possibly no B is A, therefore possibly no 
A is B: since both (propositions) 'no B is A' and 'no A is B' 
follow from one another. So if one is contingent, the other is 
too: otherwise the contingent would follow from the neces- 
sary. . . . 

33.12 If the premisses are necessary, the conclusion is 
necessary. 

With the help of these and other laws known to us from the 
chapter on propositional consequences, Pseudo-Scotus proceeds 
to establish two systems of syllogistic, one with modal propositions 
in the composite, the other in the divided sense. As premisses he 
uses not only contingent but also (one-sidedly: (15.071) possible 
and impossible propositions. We cite only a few examples from his 
teaching on conversion: 

33.13 Modal proposition in the composite sense are con- 
verted in just the same way as assertoric ones. 

33.14 Affirmative possible (de possibili) propositions in the 
divided sense (in which the subject stands) for that which is, 
are not properly speaking converted. Proof: on the supposition 
that whatever is in fact running is an ass, the following is 
true : 'every man can run' in the sense that everything which 
is a man is able to run, but its converse is false: 'a certain 
runner can be a man', . . . And I say 'properly speaking' on 
purpose, since (these propositions) can in a secondary sense 
be converted into assertorics. E.g. 'every man can run, 
therefore a certain thing that can run is a man'. . . . 

33.15 The third thesis concerns affirmative possible pro- 
positions, in which the subject stands for that which can 
exist, for such affirmatives are converted in the same way as 
assertorics. . . . 

33.16 As concerns necessary propositions, and first those 
which are to be understood in this (divided) sense with subject 
standing for what is, the first thesis is this, that affirmatives. . . 
are not converted; for supposing that God is creating, it 
does not follow: 'whatever is creating is necessarily God, 
therefore a certain God is creating necessarily'. 

33.17 The second thesis is that negative necessary propo- 
sitions with subject standing for what is, are not converted. . . . 

226 



MODAL SYLLOGISTIC 

33.18 Third thesis: that affirmative necessary propositions 
with subject standing for what can be are not properly 
speaking converted. . . . 

33.19 Fourth thesis about necessary propositions with 
subject standing for what can be : universal negatives are 
simply converted, particular negatives not. Proof: since, as 
has been said earlier, the particular affirmative possible 
(proposition with subject standing) for what can be, converts 
simply, and it contradicts the universal negative necessary 
(proposition with subject standing) for what can be; so: if 
one of two contradictories is simply converted, so is the other, 
since when the consequent follows from the antecedent, the 
opposite of the antecedent follows from the opposite of the 
consequent. 

c. OCKHAM 

Pseudo-Scotus introduced 1. one-sidedly possible premisses into 
the syllogistic, 2. in the composite sense. Ockham has a further 
innovation: he treats also of syllogisms in which one premiss is 
taken in the composite, the other in the divided sense. At the same 
time the whole modal syllogistic is formally developed from its 
structural bases with remarkable acumen. We give only two 
examples: 

33.20 As to the first figure it is to be known that when 
necessary premisses are taken in the composite sense, or 
when some are taken that are equivalent to those propositions 
in the composite sense, there is always a valid syllogism 
with a conclusion that is similar in respect of the composite or 
equivalent sense. . . . But when all the propositions are taken 
in the divided sense, or equivalent ones, a direct conclusion 
always follows, but not always an indirect. The first is evident 
because every such syllogism is regulated by did de omni or 
[did) de nullo. For by such a universal proposition it is denoted 
that of whatever the subject is said, of that the predicate is 
said. As by this: 'every man is necessarily an animal' is 
denoted that of whatever the subject 'man' is said, of that the 
predicate 'animal' is necessarily said. And the same holds 
good proportionately of the universal negative. Therefore 
adjoining a minor affirmative in which the subject (of the 
major) is predicated of something with the mode of necessity, 
the inference proceeds by did de omni or de nullo (cf. 14.23). 
Hence it follows validly: 'every man is necessarily an animal, 

227 



SCHOLASTIC LOGIC 

Socrates is necessarily a man, therefore Socrates is necessarily 
an animal'. But the indirect conclusion, viz. the converse of 
that conclusion with no other variation than the transposition 
of terms does not follow . . . (There follows here the reason as 
in 33.16). But if the major is taken in the composite sense, 
or an equivalent (proposition), and the minor in the divided 
sense, the conclusion follows in the composite sense and not in 
the divided. The first is evident, because it follows validly: 
'This is necessary: every divine person is God, one creating is 
necessarily a divine person, therefore one creating is necessa- 
rily God'. But this does not follow: 'therefore this is necessary: 
one creating is God'. 

But if the major is taken in the divided sense, or an equi- 
valent (proposition), and the minor in the composite sense, 
the conclusion follows in the divided sense and in the composite 
sense. And the reason is that it is impossible that something 
(B) essentially (per se) or accidentally inferior (to A) should 
be necessarily predicated of something (67), so that the 
proposition ('67 is B') would be necessary, without the 
proposition in which the superior (A) of that inferior (B) is 
predicated of the same (67) (viz. '67 is A') also being neces- 
sary. 

So for every Aristotelian formula Ockham has the four: (1) with 
both premisses in the composite sense, like Theophrastus (cf. 
17.15 f.); (2) with both premisses in the divided sense, like 
Aristotle - as was concluded from the indications available (§ 15, B) ; 
(3) the major premiss composite, the minor divided ; (4) the major 
premiss divided, the minor composite. 

Another example is the following treatment of syllogisms with 
both premisses in the mode of simple (one-sided) possibility: 

33.21 ... I here understand 'possible' in the sense of the 
possibility which is common to all propositions that are not 
impossible. And it is to be known that in every figure, if all 
the propositions be taken as possible in the compounded 
sense, or if equivalent ones to those be taken, the syllogism is 
invalid because inference would proceed by this rule: the 
premisses are possible, therefore the conclusion is possible, 
which rule is false. Hence it does not follow: 'that everything 
coloured is white is possible, that everything black is coloured 
is possible, therefore that everything black is white is pos- 
sible'. . . . And so the rule is false: the premisses are possible 

228 



MODAL SYLLOGISTIC 

therefore the conclusion is possible. But this rule is true: if 
the premisses are possible and compossible, the conclusion is 
possible (cf. 31.27) . . . But if the possible proposition be taken 
in the divided sense, or an equivalent one be taken, such as are 
propositions like 'every man can be white', 'a white thing can 
be black' etc. . . . there the subject can stand for things 
which are or for things which can be, i.e. for things of which it 
is verified by a verb in the present, or for things of which it is 
verified by a verb of possibility. ... As if I say: 'every white 
thing can be a man', one sense is this: everything which is 
white can be a man, and this sense is true if there be nothing 
white but man. Another sense is this: everything which can 
be white can be a man, and this is false whether only man be 
white or something other than man. . . . 

And it is to be known that if the subject of the major be 
taken for things which can be . . . however the subject of the 
minor be taken, the uniform syllogism is always valid and is 
regulated by did de omni or de nullo, and the common prin- 
ciples of the assertoric syllogism hold. E.g. if one argues thus: 
'every white thing can be a man - i.e. everything that can be 
white can be a man - every ass can be white, therefore every 
ass can be a man'. . . . 

But if the subject of the major supposes for things which 
are, then such a uniform (syllogism) is not valid, for it does not 
follow: 'everything which is white can be a man, every ass 
can be white, therefore every ass can be a man'. For if there 
be nothing white but man, the premisses are true and the 
conclusion false. . . . 

These examples may suffice to give an idea of the problems 
discussed. 

We now give a summary of the different kinds of modal syllogism 
which Ockham considered. He distinguishes the following functors 
and kinds of functor: (1) 'necessary', (2) 'possible' (one-sidedly), 
(3) 'contingent' (two-sidedly, (4) 'impossible', (5) other modes 
(subjective). Further there are (6) the assertoric propositions. Ock- 
ham deals with syllogisms with premisses in the following combi- 
nations: 



3-5 

229 



1-1 


6-1 






2-2 


6-2 


1-2 




3-3 


6-3 


1-3 


2-3 


4-4 


6-4 


1-4 


2-4 


5-5 


6-5 


1-5 


2-5 



SCHOLASTIC LOGIC 

Altogether then he has eighteen classes. In each he discusses the 
four formulae mentioned above, and this in each of three figures - the 
analogates therefore of the nineteen classical moods. Theoretically 
this gives 1368 formulae, but many of them are invalid. 

Here, however, as with Aristotle (§ 15, D), there are also many 
moods without analogues in the assertoric syllogistic, so that the 
total number of valid modal syllogisms for Ockham, in spite of the 
many invalid analogues, may reach about a thousand. 



D. LOGIC OF PROPOSITIONS IN FUTURE AND PAST TENSES 

The Scholastics did not look on propositions about the future 
and the past as modal, but they treated them quite analogously to 
modals. Two texts from Ockham illustrate this point: 

33.22 Concerning the conversion of propositions about the 
past and the future, the first thing to be known is that every 
proposition about the past and the future, in which a common 
term is subject, is to be distinguished ... in that the subject 
can suppose for what is or for what has been, if it is a proposi- 
tion about the past . . . e.g. 'the white thing was Socrates' is 
to be distinguished, since 'white' can suppose for what is 
white or for what was white. But if the proposition is about 
the future, it is to be distinguished because the subject can 
suppose for what is or for what will be. . . . Secondly it is to be 
known that when the subject of such a proposition supposes for 
what is, then the proposition should be converted into a 
proposition about the present, the subject being taken with 
the verb 'was' and the pronoun 'which', and not into a 
proposition about the past. Hence this consequence is not 
valid : 'no white thing was a man, therefore no man was white', 
if the subject of the antecedent be taken for what is. For let 
it be supposed that many men both living and dead have been 
white, and that many other things are and have been white, 
and that no man is now white, then the antecedent is true and 
the consequent false. . . . And so it should not be converted as 
aforesaid but thus: 'no white thing was a man, therefore 
nothing which was a man was white'. 

Then an example from syllogistic : 

33.23 Now we must see how syllogisms are to be made 
from propositions about the past and the future. Here it is 
to be known that when the middle term is a common term, if 
the subject of the major supposes for things which are, the 

230 



OTHER FORMULAE 

minor should be about the present and not the future or the 
past; for if the minor proposition was about the past and not 
the present such a syllogism would not be governed by did de 
omni or de nullo, because in a universal major about the past 
with subject supposing for things which are, it is not denoted that 
the predicate is affirmed or denied by the verb in the past about 
whatever the subject is affirmed of by the verb in the past. But 
it is denoted that the predicate is affirmed or denied by the verb 
in the past about whatever the subject is affirmed of by a verb 
in the present. . . . But if the subject of the major supposes for 
things which have been, then one should not adjoin a minor 
about the present, because as is quite evident, the inference 
does not proceed by did de omni or de nullo; but a minor about 
the past should be taken, and it makes no difference whether 
the subject of the minor supposes for things which are or 
things which have been. Hence this syllogism is invalid: 
'every white thing was a man, an ass is white, therefore an 
ass was a man'. . . . What has been said about propositions 
concerning the past, is to be maintained proportionately for 
those about the future. 

These principles are then applied to the syllogisms in the different 
figures. 

§34. OTHER FORMULAE 

In view of what we know about e.g. the composite and divided 
senses (29.13), and of our occasional discoveries of similar doctrines 
(28.15ff.), we must suppose that the Scholastics developed a num- 
ber of logical theories not pertaining either to propositional logic or to 
syllogistic in the Aristotelian sense. But this field is hardly at all 
explored ; e.g. we do not know whether they were acquainted with a 
more comprehensive logic of relations than that of Aristotle. 

We cite a few texts belonging to such theories, viz. (1) a series of 
texts about non-Aristotelian 'syllogisms' with singular terms, (2) an 
analysis of the quantifiers 'ever' and 'some', (3) a 'logical square' 
of so-called 'exponible' propositions, i.e. of propositions equivalent 
to the product or sum of a number of categoricals. In that connec- 
tion we finally give some theorems about the so-called syllogismus 
obliquus, which was not without importance for the later history of 
logic. 

Here it must be stressed even more than usual, that these are 
only fragments concerning a wide range of problems that has not 
been investigated. 

231 



SCHOLASTIC LOGIC 

A. SYLLOGISMS WITH SINGULAR TERMS 

A first widening of the Aristotelian syllogistic consists in the 
admission of singular terms and premisses.* Ockham already knows 
of the substitution that was to become classic : 

34.01 Every man is an animal; 
Socrates is a man; 

Therefore, Socrates is an animal. 

Here the minor premiss is singular. But Ockham also allows 
singular propositions as major premisses: 

34.02 For it follows validly (bene): 'Socrates is white, 
every man is Socrates, therefore every man is white'. . . . And 
such a syllogism ... is valid, like that which is regulated by 
did de omni or de nullo, since just as the subject of a universal 
proposition actually stands for all its significates, so too the 
singular subject stands for all its significates, since it only 
has one. 

The difference between a syllogism as instanced in 34.01 and the 
classical Aristotelian syllogism is only 'purely verbal' (34.03)! 

This may well be termed a revolutionary innovation. Not only are 
singular terms admitted, contrary to the practice of Aristotle, but 
they are formally equated with universal ones. The ground advanced 
for this remarkable position is that singular terms are names of 
classes, just like universal terms, only in this case* unit-classes. There- 
fore 34.02 is not propounding the syllogism as a substitution in the 
rule: 

34.021 If 'for all x: if x is an S then x is a P' holds, and 
l a is an S' holds, then it also holds : 'a is a P'. 

- where 'S' and 'P' are to be thought of as class-names, 'a' as an 
individual name -, but as a substitution in: 

34.022 If 'for alia;: if x is an M then x is a P' holds, and 'for 
all x: if x is an 5 then x is an M' holds, then it holds: 'for all 
x: if x is an S then x is a P' 

- where 'M', 'S' and 'P' are all class-names. In that case the sole 
difference between the Aristotelian and Ockhamist syllogisms is that 
the former is a proposition, the latter a rule. Admittedly the basis of 
the system is altered with the introduction of names for unit- 
classes. 

Again, the syllogisms with singular terms that are usually attri- 
buted to Peter Ramus, are already to be found in Ockham. 
34.02 contains one example; here is another: 

* But 34.01 is Stoic. Vid. Sextus Empiricus, Pyrr. Hyp. B 164 ff. 

232 



OTHER FORMULAE 

34.04 Although it has been said above that one cannot argue 
from affirmatives in the second figure, yet two cases are to be 
excepted from that general rule. The first is, if the middle term 
is a discrete term, for then one can infer a conclusion from 
two affirmatives, e.g. it follows validly: 'every man is Socrates, 
Plato is Socrates, therefore Plato is a man'. And such a 
syllogism can be proved, because if the propositions are con- 
verted there will result an expository syllogism in the third 
figure. 

The proof offered at the end of that text is evidently connected 
with the Aristotelian ecthesis (13.13), as is suggested by the scholastic 
term 'expository syllogism' and the following text from Ockham: 

34.05 Besides the aforesaid syllogisms, there are also 
expository syllogisms, about which we must now speak. 
Where it is to be known that an expository syllogism is one 
which is constituted by two singular premisses arranged in 
the third figure, which, however, can yield both a singular, 
and a particular or indefinite conclusion, but not a universal 
one, just as two universals in the third figure cannot yield a 
universal. ... To which it must be added that the minor 
must be affirmative, because if the minor is negative the 
syllogism is not valid. ... If the minor is affirmative, whether 
the major is affirmative or negative, the syllogism is always 
valid. 

Stephen de Monte summarizes this doctrine in systematic 
fashion : 

34.06 But it is asked whether we can rightly syllogize by 
means of an expository syllogism in every figure ; I say that we 
can. For affirmatives hold in virtue of this principle : when two 
different terms are united with some singular term taken 
singularly and univocally, in some affirmative copulative 
proposition from which the consequence holds to two uni- 
versal affirmatives (de omni), such terms should be mutually 
united in the conclusion. . . . But negatives hold in virtue of 
this principle: whenever one of two terms is united with a 
singular term etc., truly and affirmatively, and the other 
negatively, such terms should be mutually united negatively, 
respect being had to the logical properties. . . . 

Seven syllogisms arise in this way, two in each of the first and 
third figures, three in the second. 



SCHOLASTIC LOGIC 
B. ANALYSIS OF 'EVERY' AND 'SOME' 

34.07 We proceed to the signs which render (propositions) 
universal or particular. ... Of such signs, one is the universal, 
the other is the particular. The universal sign is that by which 
it is signified that the universal term to which it is adjoined 
stands copulatively for its suppositum (per modum copula- 
tionis). . . . The particular sign is that by which it is signified 
that a universal term stands disjunctively for all its supposita. 
And I purposely say 'copulatively' when speaking of the uni- 
versal sign, since if one says : 'every man runs 'it follows formal- 
ly: 'therefore this man runs, and that man runs, etc' But of 
the particular sign I have said that it signifies that a universal 
term to which it is adjoined stands disjunctively for all its 
supposita. That is evident since if one says: 'some man runs' 
it follows that Socrates or Plato runs, or Cicero runs, and so 
of each (de singulis). This would not be so if this term did 
not stand for all these (supposita); but it is true that this is 
disjunctive. Hence it is requisite and necessary for the truth 
of this: 'some man runs', that it be true of some (definite) 
man to say that he runs, i.e. that one of the singular (pro- 
positions) is true which is a part of the disjunctive (proposi- 
tion) : 'Socrates (runs) or Plato runs, and so of each', since it is 
sufficient for the truth of a disjunctive that one of its parts be 
true (cf. 31.10 and 31.223). 

This is the quite 'modern' analysis of quantified propositions 
(44.03) in the following equivalences: 

34.071 (For all x: x is F) if and only if: [a is F) and (b is F) 
and (c is F) etc. 

34.072 (There is an x such that x is F) if and only if: 
{a is F) or (b is F) or (c is F) etc. 

Further remarkable is the express appeal to a propositional rule. 
In this text propositional logic is consciously made the basis of term- 
logic, and this is only one of many examples. 

C. EXPONIBLE PROPOSITIONS 
The so-called'exponible' propositions were scholastically discussed 
in considerable detail. They are those which are equivalent to a 
product or sum of a number of categoricals. There are three kinds, 
the exclusive, the exceptive, and the reduplicative. In view of the 
metalogical treatment we give the 'logical squares' of Tarteret for the 
first two kinds, with a substitution, also from him, and the mnemonic 
expressions: 

234 



OTHER FORMULAE 



34.08 DIVES 

'Only man is an 

animal' is thus 

expounded : CONTRARY 

(1) man is an 

animal and (2) 

nothing which is 

not man is an 

animal. 



OR AT 

'Only man is 
not an animal' : 

(1) man is not 
an animal and 

(2) everything 
that is not man 
is an animal. 



(X 

w 

H 
< 

pq 

D 

C/2 



% 



■I 



-•\ 






^ 

CT 



-O 



o 



* 



> 



CO 

CO 

> 
r 

H 
M 



'Not only man is 
not an animal' : 
(1) every man is 
an animal or (2) 
something which 
is not a man is 
not an animal 

ANNO 



SUB- 
CONTRARY 



'Not only man 
is an animal' : 

(1) man is not 
an animal or 

(2) something 
which is not 
man is an animal. 

HELI 



34.09 AM ATE 



'Every man besides 
Socrates runs' : 



PECCA TA 



'Every man besides 
Socrates does not 
run': 
(1) every man who (1) every man who 

is not Socrates CONTRARY is not Socrates does 
runs and (2) not run and (2) 

Socrates is a man Socrates is a man 

and (3) Socrates and (3) Socrates 

does not run. runs. 



235 



SCHOLASTIC LOGIC 



O 4y 


'Not every man SUB- 'Not every man 
besides Socrates CONTRARY besides Socrates 


does not run' : 


runs' : (1) some 


(1) some man who 


man who is not 


is not Socrates 


Socrates does not 


runs, or (2) Socra- 


run or (2) Socrates 


tes is not a man or 


is not a man or 


(3) Socrates does 


Socrates runs. 


not run. 




IDOLES 


COM MOD. 



The originality of the formal laws by substitution in which the 
consequences shown in these squares are gained, consists in their 
being a combination of the theory of consequences (especially the 
so-called 'de Morgan' laws, cf. 31.35 f.) with the Aristotelian doctrine 
of opposition (logical square: 12.09 f.). They are all valid, and one 
can only marvel at the acumen of those logicians who knew how 
to deduce them without the aid of a formalized theory. How com- 
plicated are the processes of thought underlying the given schemata 
can be shown by one of the simplest examples, in which ANNO 
follows from D IVES. DIVES must be interpreted thus : 

(1) Some M is L, and: no not-M is L. From that there follows by 
the rule 31.222: 

(2) no not-M is L, 

and from that in turn, by the law of subalternation (24.29, cf. 32.44) : 

(3) some not-M is not L. 

Applying the rule 31.10 (cf. 31.223) one obtains : 

(4) every M is L or some not-M is not L which was to be proved. 

D. OBLIQUE SYLLOGISMS 

The Aristotelian moods with 'indirect' premisses (16.24ff.) were 
also systematically elaborated and developed by the Scholastics. 
Ockham (34.10) already knew more than a dozen formulae of this 
kind. But so far as we know, no essentially new range of problems 



236 



ANTINOMIES 

was opened up. We cite some substitutions in such moods from 
Ockham; their discovery has been quite groundlessly attributed to 
Jungius. 

34.11 It also follows validly: 'every man is an animal, 
Socrates sees a man, therefore Socrates sees an animal'. 

34.12 It follows validly: 'every man is an animal, an ass 
sees a man, therefore an ass sees an animal'. 

34.13 It follows: 'no ass belongs to man, every ass is an 
animal, therefore some animal does not belong to man'. 



§35. ANTINOMIES 

A. DEVELOPMENT 

Concerning also the search for solutions of antinomies in the Middle 
Ages insufficient knowledge is available for us to be able to survey 
the whole development here, though J. Salamucha devoted a serious 
paper to it (35.01). The connected problems seem to have been well 
known in the middle of the 13th century, but without their impor- 
tance being realized. Albert the Great merely repeats the Aristotelian 
solution of the Liar (35.02), and again Giles of Rome (in the second 
half of the 13th century) only treats this antinomy briefly and quite 
in the Aristotelian way (35.03). Peter of Spain, whose Summulae 
treat of all the problems then considered important, considers the 
fallacy of what is 'simply and in a certain respect' (under which 
heading Aristotle deals with the Liar, cf. 23.18) (35.04), but says 
nothing about antinomies. 

However, two points are worth noting about Albert the Great; he 
is the first that we find using the expression 'insoluble' (insolubile) 
which later became a technical term in this matter, and then he has 
some formulations that are new, at least in detail. This can be seen 
in a passage from his Elenchics : 

35.05 I call 'insoluble' those (propositions) which are so 
formed that whichever side of the contradictory is granted, 
the opposite follows. . . e.g. someone swears that he swears 
falsely; he swears either what is true, or not. If he swears that 
he swears falsely, and swears what is true, viz. that he swears 
falsely, nobody swears falsely in swearing what is true : there- 
fore he does not swear falsely, but it was granted that he 
does swear falsely. But if he does not swear falsely and swears 
that he swears falsely, he does not swear what is true ; there- 

237 



SCHOLASTIC LOGIC 

fore he swears falsely: because otherwise he would not swear 
what is true when he swears that he swears falsely. 

By the time of Pseudo-Scotus the subject has become a burning 
one; he cites at least one solution that diverges from his own (35.06) 
and treats the question in two chapters of which the first bears the 
title 'Whether a universal term can stand for the whole proposition 
of which it is a part' (35.07). The answer is a decisive negative : 

35.08 It is to be said that a part as part cannot stand for 
the whole proposition. 

His solution, however, does not consist in an application of this 
principle, but is found in the distinction between the signified and the 
exercised act: 

35.09 If it is said: 'I say what is false, therefore it is true 
that I say what is false', I answer that the consequence does 
not hold formally, as (also) it does not follow: 'man is an ani- 
mal, therefore it is true to say that man is an animal', although 
the consequent is contained in the antecedent in the exercised 
act. Granted further that it follows, though not formally, I 
say that this other does not follow: 'I say that I say what is 
false, therefore in what I say I am simply truthful', or only 
in a certain respect and not simply. . . . Similarly it follows in 
some cases: 'What I say is true, therefore I am simply truth- 
ful', as (e.g.) here : 'It is true that I say that man is an animal, 
therefore I am simply truthful', viz. in those cases in which 
there is truth both in the act signified and the act exercised. 
But in our case (in proposito) there is falsity in the act signified 
and truth in the act exercised. It follows then : 'It is true that I 
exercise the act of speaking about what is false; therefore that 
about which I exercise it is false'. 

A comparison of this text with 27.13 shows that we have here 
almost exactly the modern distinction between use and mention. But 
Pseudo-Scotus, employing the same terminology, teaches just the 
opposite to Burleigh. 

These two examples are enough to show the state of affairs in the 
13th century. When we come to Ockham the antinomies are no 
longer dealt with in sophistics, but in a special chapter About 
Insolubles (35.10). After that such a treatise becomes an essential 
part of scholastic logic. We pass over the further stages of develop- 
ment, which are mostly not known, and show how far the matter 
had got by the time of Paul of Venice at the end of the Middle Ages. 

238 



ANTINOMIES 

B. FORMULATION 
1. The Liar 

35.11 I compose the much-disputed insoluble by positing 
(1) that Socrates utters this proposition: 'Socrates says what 
is false', and this proposition is A, and (2) (that he) utters no 
other (proposition besides A), (where the proposition A) (3) 
signifies so exactly and adequately that it must not be varied 
in the present reply. That posited, I submits and ask whether 
it is true or false. If it is said that it is true, contrariwise : it is 
consistent with the whole case that there is no other Socrates 
but this Socrates, and that posited, it follows that A is false. 
But if it is said that A is false, contrariwise: it is consistent 
with the whole case that there are two Socrateses of which 
the first says A, and the second that there is no God: if that 
is taken with the statement of the case, it follows that A is true. 

35.12 I suppose therefore that Socrates, who is every 
Socrates, utters this and no other proposition: 'Socrates says 
what is false', which exactly and adequately signifies (what it 
says) ; let it be A. Which being supposed, it follows from what 
has been said that A is false; and Socrates says A, therefore 
Socrates says what is false. This consequence is valid, and the 
antecedent is true, therefore also the consequent; but the 
antecedent is A, therefore A is true. 

Secondly it is argued: What is false is said by Socrates, 
therefore Socrates says what is false. The consequence holds 
from the passive to its active. But the antecedent is true, 
therefore also the consequent, and the antecedent is A, 
therefore A is true. Since, however, the antecedent is true, it is 
evident that its adequate significate is true. But it is a con- 
tradiction that it should be true. 

Thirdly it is argued: the contradictory opposite of A is 
false, therefore A is true. The consequence holds and the 
antecedent is proved: for this: 'no Socrates says what is 
false' is false, and this is the contradictory opposite of A ; 
therefore the contradictory opposite of A is false. The conse- 
quence and the minor premiss hold, and I prove the major: 
Since, A is false; but a certain Socrates says A; therefore a 
certain Socrates says what is false. Or thus: No Socrates says 
what is false; therefore no Socrates says the false .4.. The 
consequence holds from the negative distributed super- 
ordinate to its subordinate. The consequent is false, therefore 
also the antecedent. 

239 



SCHOLASTIC LOGIC 

2. Other antinomies 

Besides this 'famous' insoluble there is a long series of similar 
antinomies that derive from it, of which we give some examples 
from Paul of Venice, omitting the always recurring words 'Socrates 
who is all Socrates' and 'which signify exactly as the terms suggest 
(pretendunt)' : 

35.13 Socrates . . . believes this: 'Socrates is deceived' . . . 
and no other (proposition). 

35.14 Socrates believes this and no other: 'Plato is deceiv- 
ed' .. . but Plato . . . believes this: 'Socrates is not deceived'. 

35.15 Socrates . . . says this and nothing else: 'Socrates 
lies'. 

35.16 'Socrates is sick'; 'Plato answers falsely (male)'; 
'Socrates will have no penny'; ('Socrates will not cross the 
bridge') ;* where it is supposed that every sick man, and only 
one such, says what is false, and that every well man, and 
only one such, says what is true (and correspondingly for the 
three other cases). . . . On these suppositions I assert that 
Socrates . . . utters only the following: 'Socrates is sick' etc. 

Those are the so-called 'singular insolubles'. There follow on them 
the 'quantified' ones: 

35.17 I posit the case that this proposition 'it is false' is 
every proposition. 

35.18 Let this be the case, that there are only two proposi- 
tions, A and B, A false, and B this: 'A is all that is true'. 

35.19 I posit that A, B and C are all the propositions, 
where A and B are true, and C is this: 'every proposition is 
unlike this' indicating A and B. 

35.20 I posit that A and B are all the propositions, where 
A is this : 'the chimera exists' . . . and B this : 'every proposi- 
tion is false'. 

35.21 Let A, B and C be all the propositions . . . where A is 
this: 'God exists', Bthis: 'man is an ass', C** this: 'there are 
as many true as false propositions'. 

35.22 The answer to be given would be similar on the 
supposition that there were only five propositions ... of 
which two were true, two false, and the fifth was: 'there are 
more false than true (propositions)'. 

* Inserted according to the words just following. 
* * The text has D. 

240 



ANTINOMIES 

Then some 'exponible' insolubles: 

35.23 I posit that 'this is the only exclusive proposition' 
is the only exclusive (proposition). . . . 

35.24 Let this be a fallacy about exceptives: 'no proposition 
besides A is false', supposing that this is A, and that it is 
every proposition. 

35.25 I posit that A, B, and C are all the propositions . . . 
that A and B are true, and that C is this exclusive: 'every 
proposition besides the exclusive is true'. 

35.26 The answer is similar ... on the supposition that 
every man besides Socrates says: 'God exists', and that 
Socrates says only this: 'every man besides me says what is 
true'. 

These are only a few examples from the rich store of late scholastic 
sophistic. 

C. SOLUTIONS 
1. The first twelve solutions 

35.27 The first opinion states that the insoluble is to be 
solved by reference to the fallacy of the form of speech (11.19). 
. . . And if it is argued: 'Socrates utters this falsehood, there- 
fore Socrates says what is false', one denies the consequence 
and says: 'This is the fallacy of the form of speech, because 
by reason of the (reference of the) speech the term 'false' 
supposes for 'Socrates etc' in the antecedent, but for some- 
thing else in the consequent. . . . 

35.28 The second opinion solves the insolubles by the 
fallacy of false cause (11.24) . . . since the antecedent seems 
to be the cause of the consequent but is not. . . . 

35.29 The third opinion says that when Socrates says 
'Socrates says what is false', the word 'says', although in the 
present tense, ought to be understood for the time of the 
instant immediately preceding the time of utterance. There- 
fore it denies it (the proposition), saying that it is false. And 
then to the argument: 'this is false and Socrates says it, 
therefore Socrates says a false (proposition)', they say that 
the verb 'says' is verified for different times in the antecedent 
and consequent. . . . 

35.30 The fourth opinion states that nobody can say that he 
says what is false or understand that he understands what is 
false, nor can there be any proposition on which an insoluble 
can be based. This opinion is repugnant to sense and thought. 

241 



SCHOLASTIC LOGIC 

For everybody knows that a man can open his mouth and 
form these utterances: 'I say what is false' or sit down and 
read similar ones. . . . 

35.31 The fifth opinion states that when Socrates says 
that he himself says what is false, he says nothing. . . . This 
opinion is likewise false because in so saying, Socrates says 
letters, syllables, dictions and orations as I have elsewhere 
shown. Further Socrates is heard to speak, therefore he says 
something. Again they would have to say that if this, and no 
other, proposition was written : 'it is false', that nothing would 
be written, which is evidently impossible. 

The fifth opinion counts the insoluble as deprived of sense. 

35.32 The sixth opinion states that the insoluble is neither 
true nor false but something intermediate, indifferent to 
each. They are wrong too, because every proposition is true 
or false, and every insoluble is a proposition, therefore every 
insoluble is true or false. . . . 

That is an effort to solve the antinomy in a three-valued logic. 

35.33 The seventh opinion states that the insoluble is to 
be solved by reference to the fallacy of equivocation. For when 
it is said: 'Socrates says what is false' they distinguish about 
the 'saying' according to an equivocation: for it can signify 
saying that is exercised or that is thought (conceptum). And 
by 'saying that is exercised' is meant that which is in course of 
accomplishment; it expresses the judgment and is not com- 
pletely a dictum. But by 'saying that is thought' is meant 
(what happens) when a man has said something or spoken in 
some way and immediately after he says that he says that, 
or speaks in that way. E.g. supposes that Socrates says 
'God exists' and immediately after: 'Socrates says what is 
true'. This opinion says that when Socrates begins to say 
'Socrates says what is false', if 'saying' be taken for exercised 
saying, it is true; but if for saying in thought, it is false. And 
if it is argued: 'nothing false is said by Socrates; and this is 
said by Socrates; therefore it is not false' - they say that the 
major is verified for saying in thought, and the minor for 
exercised saying, and so (the argument) does not conclude. 
But this solution is no use, for let it be supposed that the 
speech is made with exercised saying, and the usual deduction 
will go through. . . . 

242 



ANTINOMIES 

This solution corresponds with that of Pseudo-Scotus above 
(35.09). 

35.34 The eighth opinion states that no insoluble is true or 
false because nothing such is a proposition. For although 
every or any insoluble be an indicative statement signifying 
according as its signification is or is not, yet this is not 
sufficient for it to be called a 'proposition'. Against this 
opinion it is argued that it follows from it that there are some 
two enunciations of which the adequate significate is one and 
the same, yet one is a proposition, the other not, as is clear 
when one supposes these: 'this is false' and 'this is false', 
indicating in both cases the second of them. . . . 

This is again a quite 'modern' conception. Paul of Venice, and. it 
would seem, the majority of late Scholastics, did not like it. 

35.35 The ninth opinion states that the insoluble is true or 
false, but not true and not false. . . . 

Here the alternative l A is true or false' seems to be admitted, but 
'A is true' and 'A is false' to be both rejected. 

35.36 The tenth opinion solves the insoluble by reference 
to the fallacy of in a certain respect and simply (11.24), 
saying that an insoluble is a difficult paradox (paralogismus) 
arising from (a confusion between what is) in a certain respect 
and simply, due to the reflection of some act upon itself with 
a privative or negative qualification. So in solving, it says that 
this consequence is not valid : 'this false thing is said by 
Socrates, therefore a false thing is said by Socrates', supposing 
that Socrates says the consequent and not something else 
which is not part of it - because the argument proceeds from 
a certain respect to what is simply so ; for the antecedent only 
signifies categorically, but the consequent hypothetically, 
since it signifies that it is true and that it is false. . . . 

35.37 The eleventh opinion, favouring the opinion just 
expounded, states that every insoluble proposition signifies 
that it is true and that it is false, when understood as referring 
to its adequate significate. For, as is said, every categorical 
proposition signifies that that for which the subject and 
predicate suppose is or is not the same thing, and the being or 
not being the same thing is for the proposition, affirmative or 
negative, to be true; therefore every categorical proposition, 
whether affirmative or negative, signifies that itself is true, 

243 



SCHOLASTIC LOGIC 

and every insoluble proposition falsifies itself; therefore every 
insoluble proposition signifies that it is true or that it is 
false. . . . 

The last two opinions consider the insoluble to be equivalent to a 
copulative proposition. Why it should be so we shall see below (35.44) . 

35.38 The twelfth opinion, commonly held by all today, is 
that an insoluble proposition is a proposition which is supposed 
to be mentioned, and which, when it signifies precisely accord- 
ing to the circumstances supposed, yields the result that it is 
true and that it is false. E.g. if a case be posited about an 
insoluble, and it is not posited how that insoluble should 
signify, it is to be answered as though outside time: e.g. if 
it be supposed that Socrates says: 'Socrates says what is 
false' without further determination, the proposition advanc- 
ed: 'Socrates says what is false' is to be doubted. But if it 
be supposed that the insoluble signifies as the terms suggest, 
the supposition is admitted and the insoluble is granted, and 
one says that it is false. And if it be said : 'this is false : "Socra- 
tes says what is false", therefore it signifies as it is not, but 
signifies that Socrates says what is false, therefore etc' - the 
consequence is denied. But in the minor it should be added 
that it signifies precisely so, and if that is posited, every 
such supposition is denied. . . . 

The 'time of obligation' here referred to is a technical term of 
scholastic discussion (tractatus de obligationibus: cf. § 26, D), on 
which very little research has so far been done. It means the time 
during which the disputant is bound to some (usually arbitrary) 
supposition. 

2. The thirteenth solution 

35.39 The thirteenth opinion states a number of conjuncts, 
some in the form of theses (conclusionum), others in the form 
of suppositions, others in the form of propositions or corol- 
laries ; but all these can be briefly stated in the form of theses 
and corollaries. 

35.40 The first thesis is this : no created thing can distinctly 
represent itself formally, though it can do so objectively. 
This is clear, since no created thing can be the proper and 
distinct formal cognition of itself; for if something was to be 
so, anything would be so, since there would be no more 
reason in one case than in another. E.g. we say that the 

244 



ANTINOMIES 

king's image signifies the king not formally but objectively, 
while the mental concept which we have of the king signifies 
the king not objectively but formally, because it is the formal 
cognition of the king. But if it be said that it represents itself 
distinctly, this will be objectively, by another concept 
(noiilia) and not formally, by itself. 

35.41 Second thesis: no mental proposition properly so- 
called can signify that itself is true or that itself is false. 
Proof: because otherwise it would follow that some proper 
and distinct cognition would be a formal cognition of itself, 
which is against the first thesis. 

From this thesis it follows that the understanding cannot 
form a universal mental proposition properly so-called which 
signifies that every mental proposition is false, such as this 
mental (proposition): 'every mental proposition is false', 
understanding the subject to suppose for itself; nor can it 
form any mental proposition properly so-called which signifies 
that any other is false which in turn signifies that the one 
indicated by the first is false; nor any mental proposition 
properly so-called which signifies that its contradictory is 
true, as this one: 'this is true' indicating its contradictory. . . . 

The last two texts contain a notably acute formulation of the 
veto on circulus vitiosus (48.21), and so of the most important modern 
idea about the solution of the antinomies. 

35.42 The third thesis is this : a part of a mental proposition 
properly so-called cannot suppose for that same proposition 
of which it is a part, nor for the contradictory of that proposi- 
tion; nor can a part of a proposition that signifies in an 
arbitrary way suppose for the corresponding mental proposi- 
tion properly so-called. From which it follows that if this 
mental proposition is formed, and no other: 'every mental 
proposition is universal', it would be false. 

35.43 Fourth thesis: there might be a vocal or written or 
mental proposition improperly so-called which had reflection 
on itself, because all such signify in an arbitrary way and not 
naturally, objectively but not formally. But a mental proposi- 
tion properly so-called is a sign that represents naturally and 
formally, and it is not in our power that such a sign should 
signify whatever we want, as it is in the case of a vocal, 
written or mental sign improperly so-called. 

From this thesis it follows that every insoluble proposition 

245 



SCHOLASTIC LOGIC 

is a vocal, written or mental proposition improperly so-called; 
and a part of any such can suppose for the whole of which it 
is a part. 

35.44 The fifth thesis is this : to every insoluble proposition 
there corresponds a true mental proposition properly so- 
called, and another one properly so-called, false. This is 
evident in the following: 'this is false' indicating itself, which 
corresponds to one such mental proposition, 'this is false', 
which is true. And the second part is proved. For this vocal 
proposition is false, therefore it signifies that a mental one is 
false, but not the one expressed, therefore another one which 
is true, viz. 'this is false', indicating the first mental one which 
indicates a vocal or written one. 

35.45 From this thesis there follow some corollaries. 
First, that any insoluble proposition, and its contradictory too, 
is a manifold proposition (propositio plures) because there 
correspond to it a number of distinct (inconiundae) mental 
propositions. 

Second, there are some propositions, vocally quite similar 
and with terms supposing for the same things, one of which 
is a manifold proposition, but not the other. This is clear in the 
following: 'this is false' and 'this is false' where each 'this' 
indicates the second proposition. 

Third corollary, every insoluble proposition is simultaneous- 
ly true and false, and its contradictory likewise, because two 
mental propositions of which one is true and the other false 
contradict one another, though neither is simply true or 
simply false, but in a certain respect. . . . 

3. The fourteenth solution 

35.46 The fourteenth opinion, which is the basis of many 
of the preceding ones and so of those disputants who try 
rather to evade (the difficulties) than to answer, states that 
the insolubles are to be solved by means of the fallacy of the 
accident, according to which paradoxes (paralogismi) arise 
in two ways, by variation of the middle term or of one of the 
extremes. By variation of the middle, as when the middle 
supposes for something different in the major to what it 
supposes for in the minor, and conversely. And similarly 
when an extreme is varied. This opinion therefore says that 
when Socrates says 'Socrates says what is false', he says what 
is false. And then in reply to the argument: 'Socrates says 

246 



ANTINOMIES 

this, and this is false, therefore Socrates says what is false' 
they deny the consequence, saying that here is a fallacy of the 
accident due to variation in an extreme; for the term 'false' 
supposes for something in the minor for which it does not 
suppose in the conclusion. Similarly if it is argued from the 
opposite saying of Socrates: 'nothing false is said by Socrates; 
this is false; therefore this is not said by Socrates', this is a 
fallacy of the accident due to variation of the middle ; for the 
term 'false' supposes for something in the major for which it 
does not suppose in the minor. 

To show that, they presuppose that in no proposition does 
a part suppose for the whole of which it is a part, nor is it 
convertible with the whole, nor antecedent to the whole. 
From which it is clear that the proposition 'Socrates says 
what is false, signifies that Socrates says what is false, not, 
however, the false thing that he says, but some false thing 
distinct from that; but because he only says that proposition, 
therefore it is false. . . . 

This opinion has been met with already in Pseudo-Scotus (35.06; 
and is adopted by others too. 

4. Preliminaries to the solution of Paul of Venice 

After expounding fourteen opinions none of which are acceptable 
to him, Paul of Venice gives his own solution, and takes occasion to 
collect the current late-scholastic teachings relevant to the antino- 
mies. We reproduce the essentials. 

35.47 To explain the fifteenth opinion, which I know to 
be that of good (logicians) of old times, three chapters (articuli) 
are adduced. The first contains an explanation of terms, the 
second introductory suppositions, the third our purpose in the 
form of theses. 

35.48 As to the first, this is the first division : every insoluble 
arises either from our activity or from a property of the 
expression (vocis). Our acts are twofold, some interior, others 
exterior. Interior are such as imagining, thinking etc. ; exterior 
are bodily ones such as saying, speaking etc. Insolubles arising 
from our activity are: 'Socrates says what is false', 'Socrates 
understands what is false' etc. Properties of the expression 
are such as being subject, having appellation, being true or 
false, being able to be true, not being true of something other 
than itself, and so simply (de se) false, and not being false of 
itself or of something else. And so there arise from properties 

247 



SCHOLASTIC LOGIC 

of the expression insolubles like these: 'it is false', 'nothing 
is true', 'the proposition is not verified of itself. . . 

35.49 The second division is this: some propositions have 
reflection on themselves, some do not. A proposition having 
reflection on itself is one whose signification reflects on itself, 
e.g. 'it is every complex thing', or 'this is false', indicating 
itself. A proposition without reflection on itself is one whose 
significate is not referred to itself, e.g. 'God exists' and 'man 
is an ass'. 

35.50 The third division is this: of propositions having 
reflection on themselves some have this reflection immediately, 
others have it mediately. . . . 

35.51 The fourth division: of propositions having reflec- 
tions on themselves, some have the property that their 
significations terminate solely at themselves, e.g. 'this is 
true', 'this is false', indicating themselves. But others have 
the property that their significations terminate both at 
themselves and at other things, e.g. 'every proposition is 
true', 'every proposition is false'. For they do not only signify 
that they alone are true or false, but that other propositions 
distinct from them are so too. . . . 

35.52 It follows that no proposition has reflection on itself 
unless it contains a term that is appropriated to signify the 
proposition, such as are the terms 'true', 'false', 'universal', 
'particular', 'affirmative', 'negative', 'to be granted', 'to be 
denied', 'to be doubted' and so on. But not every proposition 
containing such a term has to have reflection on itself, as is 
clear in these cases: 'it is false', when this is true, and again 
'this is true' indicating 'God exists'; for such does not have 
reflection on itself, but its signification is directed solely to 
what is indicated etc. . . . 

There again is the 'modern' notion of the vicious circle (48.21). 
Here are a few more preliminaries : 

35.53 The first introductory supposition is this: that that 
proposition is true whose adequate significate is true, and if 
its being true contains no contradiction. . . . 

35.54 Second supposition: that proposition is said to be 
false which falsifies itself or whose falsity arises not from the 
terms but from its false adequate significate. From which it 
follows that there is a false proposition with a true adequate 
significate, as is clear in the following: 'that is false', indicating 

248 



ANTINOMIES 

itself. That it is false is evident, since it states that it is false, 
therefore it is false; and so its adequate significate is true, since 
it is true that it (the proposition) is false. It follows that 
every proposition which falsifies itself is false, and that not 
every proposition which verifies itself is true ; since this : 
'every proposition is true' verifies itself but is not true, as is 
evident. 

35.55 The third supposition is this: two propositions are 
equivalent (invicem convertuntur) if their adequate significates 
are identical. For let A and B be two such propositions having 
the same adequate significate, and I argue thus: A and B 
have all extremes the same, vocally and in writing, and in 
thought, and similar copulas, and there is no indication 
belonging to one which does not belong to the other; then 
they are equivalent. 

There follow some further preliminary suppositions taken from 
the generally received teaching about supposition and consequence. 
Finally this: 

35.56 The last supposition is this: a part of a proposition 
can stand for the whole of which it is a part, as also for 
everything which belongs to it, without restriction, whether 
in thought or in writing or in speech. 

Thereby is rejected the thirteenth opinion (35.39 ff.), and with 
it the modern principle according to which an insoluble is not a 
proposition since it contains a part standing for the whole (48.12 f.). 
This principle seems to be presupposed in various ways by the 
fourth (35.30), fifth (35.31), eighth (35.34), tenth (35.36)," and 
eleventh (35.37) opinions. 

The rejection of the thirteenth opinion means that the current 
modern distinction of language and meta-language was not adopted 
by Paul of Venice for his own solution. But it is explicitly accepted 
in the fifth thesis of the thirteenth opinion (35.44), more or less so 
in some of the other opinions. 

5. The solution of Paul of Venice 

Paul's own solution is very like that of the eleventh (35.37) and 
twelfth (35.38) opinions, and so we do not reproduce his long and 
difficult text. It consists essentially in a sharp distinction between 
the ordinary and 'exact and adequate' meaning of the insoluble 
proposition, where 'exact and adequate' connotes: 

(1) the semantic correlate, that to which it refers; 

(2) that the proposition itself is true. 

249 



SCHOLASTIC LOGIC 

This was said already in 35.57, though without the use of 'exact 
and adequate', and with a universality that Paul does not approve. 
We repeat the main ideas, as they underlie his own solution. Some 
simplification will be effected, and formalization used. The first 
thing is to set out the antinomy, for which four extralogical axioms 
are employed : 

(1)^4 signifies: A is false. 

(2) If A signifies p, then A is true if and only if p. 

(3) If A signifies p, then A is false if and only if not-p. 

(4) A is false if and only if A is not true. (1) is the 'insoluble' 
proposition itself, (2)-(4) are various formulations of the Aristotelian 
definitions of truth and falsity (10.35). Substituting l A is false' 
for 'p' in (2), we get by (1): 

(5) A is true if and only if A is false, which with (4) gives : 

(6) A is true if and only if A is not true, which in turn yields : 

(7) A is not true, 
and so by (4) : 

(8) A is false. 

But if we put l A is false' for 'p' in (3), we get: 

(9) A is false if and only if A is not false, which immediately 
yields: 

(10) A is not false, 

in contradiction to (8). Here then is a genuine antinomy. 

But the antinomy does not emerge if we operate with the 'exact 
and adequate' meaning instead of the simple one. (1) and (4) remain, 
but the other two axioms take on these forms : 

(2') If A signifies p, then A is true if and only if [(1) A is true, and 
(2) p]. 

(3') If A signifies p, then A is false if and only if not [(1) A is true, 
and (2) p], since as has been said, a proposition has 'exact and 
adequate' signification when it signifies that it is itself true, and that 
what it states is as it is stated to be. 

The first part of the deduction now goes through analogously to 
that given above, and we again reach: 

(8') A is false. 
But putting 'A is false' for 'p' in (3') gives: 

(9') A is false if and only if not [(1) A is true, and (2) A is false], 
to which we can apply the de Morgan laws (cf. 31.35) to get: 

(10') A is false if and only if either (1) A is not true, or (2) A is not 
false, i.e. in view of (4): 

(IT) A is false if and only if either (1) A is false or (2) A is not 
false. 

As that alternation is logically true, being a substitution in the 
law of excluded middle (cf. 31.35), the first part of the equivalence 
must also be true, giving us: 

(12') A is false 

250 



SUMMARY 

which so far from being in contradiction to (8') is equiform with it. 
The antinomy is solved. 

So far as we know, the medieval logicians only treated of semantical, 
not of logical antinomies. But the solutions contain all that is required 

for those as well. 



SUMMARY 

In summary, we can make the following statements about medie- 
val formal logic, in spite of our fragmentary knowledge: 

1. Scholasticism created a quite new variety of formal logic. The 
essential difference between this and the one we found among the 
ancients, consists in its being an endeavour to abstract the laws and 
rules of a living (Latin) language, regard had to the whole realm of 
semantical and syntactical functions of signs. 

2. This endeavour led to the codification of a far-reaching and 
thorough semantics and syntax; semiotic problems hold the fore- 
front of interest, and nearly all problems are treated in relation to 
them. 

3. Hence this logic is nearly entirely conducted in a meta-language 
(§ 26, B) with a clear distinction between rules and laws. Most of the 
theorems are thought of as rules and formulated descriptively. 

4. The problem of logical form (§ 26, C) is posed and solved with 
great acumen. 

5. Problems of propositional logic and technique are investigated 
as thoroughly and in as abstract a way as anywhere among the 
Megarian-Stoics. 

6. Asserloric term-logic consists here essentially in a re-interpreta- 
tion and acute development of syllogistic. But there are also other 
kinds of problem in evidence, such as that of plural quantification, of 
the null class, perhaps of relation-logic, etc. 

7. Modal logic, both of propositions and terms, became one of the 
most important fields of investigation. Not only was the traditional 
system analysed with amazing thoroughness, but quite new prob- 
lems were posed and solved, especially in the propositional domain. 

8. Finally the problem of semantical antinomies w r as faced in really 
enormous treatises. Numerous antinomies of this kind were posited, 
and we have seen more than a dozen different solutions attempted. 
Between them they contain nearly every essential feature of what we 
know today on this subject. 

So even in the present incomplete state of knowledge, we can 
state with safety that in scholastic formal logic we are confronted 
with a very original and very fine variety of logic. 



251 



PART IV 

Transitional Period 



§36. THE 'CLASSICAL' LOGIC 

It is usual to put the close of the medieval period of history at the 
end of the 15th century. Of course that does not mean that typically 
scholastic ways of thought did not persist longer; indeed very im- 
portant scholastic schools arose in the 16th and 17th centuries and 
accomplished deep and original investigations — it is enough to 
mention Cajetan and Vittoria. But there was no more research into 
formal logic; at most we find summaries of earlier results. 

Instead there slowly grew up something quite new, the so-called 
'classical' logic. Within this extensive movement which held the 
field in hundreds of books of logic for nearly four hundred years, one 
can distinguish three different tendencies: (1) humanism (inclusive 
of those later 17th century thinkers who were humanist in their 
approach to logic); it is purely negative, a mere rejection of Scho- 
lasticism; (2) 'classical' logic in the narrower sense; (3) more recent 
endeavours to broaden the bounds of (2). Typical examples of the 
three are L. Valla and Peter Ramus, the Logique du Port Royal, 
and W. Hamilton. 

In what follows we shall first quote some passages to illustrate 
the general attitude of authors of books entitled 'Logic' in this 
period, then some that contain contributions to logical questions, 
whether scholastic or mathematico-logical, though these contribu- 
tions are of small historical importance. 



A. HUMANISM 

Interest centres much more on rhetorical, psychological and 
epistemological problems than on logical ones. The humanists, and 
many 'classical' logicians after them, expressly reject all formalism. 
That they did not at the same time reject logic entirely is due to 
their superstitious reverence for all ancient thinkers, Aristotle in- 
cluded. But everything medieval was looked on as sheer barbarism, 
especially if connected with formal logic. Here is an instance. Valla 
writes : 

36.01 I am often in doubt about many authors of the dia- 
lectical art, whether to accuse them of ignorance, vanity or 
malice, or all at once. For when I consider the numerous 
errors by which they have deceived themselves no less than 
others, I ascribe them to negligence or human weakness. But 
when on the other hand I see that everything they have 
transmitted to us in endless books has been given in quite a 
few rules, what other reason can I suppose than sheer pride? 
In amusing themselves by letting the branches of the vine 

254 



H U M A N I S M 

spread far and wide, they have changed the true vine into a 
wild one. And when - this is the worst - I see the sophisms, 
quibbles and misrepresentations which they use and teach, 
I can only kindle against them as against people who teach the 
art of piracy rather than navigation, or to express myself 
more mildly, knowledge of wrestling instead of war. 

And again, about the third figure of the syllogism: 

36.02 trifling Polyphemus! peripatetic family, that 
loves trifles! vile people, whoever have you heard arguing 
like this? Indeed, who among you has ever presumed to 
argue so? Who permitted, endured, understood one who 
argued thus? 

It is on these 'grounds' that the third figure is to be invalid! 
In another manner, but the content goes deeper, Descartes ex- 
presses himself: 

36.03 We leave out of account all the prescription of the 
dialecticians by which they think to rule human reason, 
prescribing certain forms of discourse which conclude so 
necessarily that in relying on them the reason, although to 
some extent on holiday from the informative and attentive 
consideration (of the object), can yet draw some certain 
conclusion by means of the form. 

Evidently such an attitude will not bring one to any logic. Peter 
Ramus holds a special position among the humanists. Though, at 
least in his first period, he was perhaps the most radical anti- 
Aristotelian, yet he succeeded in formulating on occasion some 
interesting thoughts, and published extensive treatises on formal 
logic. However, the following gives some idea of the general level 
of his logic: 

36.04 Moreover, two further connected (i.e. conditional) 
moods were added by Theophrastus and Eudemus, in which 
the antecedent is negative and the consequent affirmative. 
The third connected mood, then, takes the contradictory of 
the antecedent and concludes to the contradictory (of the 
consequent), e.g. 

If the Trojans have come to Italy without due permission, 
they will be punished; 
but they came with permission, 
therefore they will not be punished. . . . 

255 



TRANSITIONAL PERIOD 

36.05 The fourth connected mood takes the consequent 
and concludes to the antecedent: . . . 

If nothing bad had happened, they would be here already; 

but they are here, 

therefore nothing bad has happened. 

36.06 This mood is the rarest of all, but natural and useful, 
strict and correct, and it never produces a false conclusion 
from true premisses. ... * 

Of course Theophrastus taught no such moods; both theorems 
are formally invalid and hold only by reason of the matter in parti- 
cular cases. It is instructive to compare these thoughts with the 
treatment of similar problems in the Stoics (22.04 f.) and Scholastics 

(§31). 

B. CONTENT 

In so bad a milieu logic could not last long, yet there were some 
thinkers among the humanists, Melanchthon for instance, who without 
being creative logicians, had a good knowledge of Aristotle. It was 
through them that in the 17th century the form of logic developed 
which we call the 'classical' in the narrower sense, partly among 
the so-called Protestant Scholastics, partly in Cartesian circles. 
Perhaps the most important representative work is the Logique 
ou Varl de penser of P. Nicole and A. Arnault. We describe the 
contents of this work, since they give the best survey of the problems 
considered in 'classical' logic. 

The book has four parts, about ideas, judgments, arguments, and 
method. In the first part the Aristotelian categories (ch. 3) and 
predicables (ch. 7) are briefly considered, along with some points of 
semantics (Des idees des choses el des idees des signes, ch. 2), com- 
prehension and extension (ch. 6). The other eleven chapters are 
devoted to epistemological reflections. 

The second part roughly corresponds to the content of Aristotle's 
Hermeneia, and includes also considerations on definition and 
division (ch. 15-16). 

In the third part the authors expound categorical syllogistic, 
apparently following Peter of Spain and so as a set of rules, but with 
use of singular premisses in the manner of Ockham (34.01 f.). Four 
figures are recognized, with nineteen moods (the subaltern moods 
are missing). There follows a chapter on hypothetical syllogisms, 
and (ch. 12) a theory of syllogismes conjonctifs (the Stoic compounds) 
using formulas of term-logic, e.g. 



* Admittedly these moods, as Professor A. Church has stated, do not appear 
in all editions. Perhaps Ramus saw his own mistake. 

256 



PSYCHOLOGISM 

36.07 If there is a God, one must love him: 
But there is a God ; 

Therefore one must love him. 

Then there are some considerations about dialectical loci. 

When we compare this with scholastic logic, the main things 
missing are the doctrines of supposition, consequences, antinomies, 
and modal logic. The main topics covered are those of the Categories, 
Hermeneia, and the first seven chapters of the first hook of the 
Prior Analytics, but the treatment is often scholastic rather than 
Aristotelian, for which we instance the use of the mnemonics 
Barbara, Celarenl etc. and the metalogical method of exposition. 

The Logica Hamburgensis of J. Jungius (1635) is much better, 
and richer in content; but it did not succeed in becoming established. 
The Logique ou Vart de penser, also called the Port Royal Logic, 
became the standard text-book, a kind of Summulae of 'classical' 
logic. All other text-books mainly repeated its contents. 

C. PSYCHOLOGISM 

'Classical' logic is characterized not only by its poverty of con- 
tent but also by its radical psychologism. Jungius provides a good 
example: 

36.08 1. Logic is the art of distinguishing truth from falsity 
in the operations of our mind (mentis). 

2. There are three operations of the mind : notion or concept, 
enunciation, and dianoea or discourse. 

3. Notion is the first operation of our mind, in which we 
express something by an image; in other words a notion is 
a simulacrum by which we represent things in the mind. . . . 

5. Enunciation is the second operation of the mind, so 
compounded of notions as to bring about truth or falsity. 
E.g. these are true enunciations: the sun shines, man is a 
biped, the oak is a tree. . . . 

9. It is to be noted that a notion and the formation of a 
notion, an enunciation and the effecting of an enunciation, 
an argumentation and the construction of an argumentation 
are one and the same. 

This is admittedly an extreme case. But when one thinks that 
the text is from Jungius, one of the best logicians of the 17th 
century, one cannot but marvel at the extent to which the under- 
standing of logic has disappeared. Even Boole will maintain much 
this idea of logic. 

257 



TRANSITIONAL PERIOD 

Poor in content, devoid of all deep problems, permeated with a 
whole lot of non-logical philosophical ideas, psychologist in the 
worst sense, - that is now we have to sum up the 'classical' logic: 

It may, however, be remarked that A. Menne (Logik und Existenz, 
131, note 34) has propounded a distinction between an at least 
relatively pure 'classical' logic, and a 'traditional' philosophical and 
psychological logic, though these terms are commonly used syno- 
nymously. Of the former, J. N. Keynes (1906) may be taken as in 
every sense the best representative, but W. E. Johnson (1921) 
shows how relative is the distinction. 

D. LEIBNIZ 

Formed by this logic and its prejudices, modern philosophers such 
as Spinoza, the British empiricists, Wolff, Kant, Hegel etc. could 
have no interest for the historian of formal logic. When compared 
with the logicians of the 4th century B.C., the 13th and 20th cen- 
turies a.d. they were simply ignorant of what pertains to logic and 
for the most part only knew what they found in the Port Royal 
Logic. 

But there is one exception, Leibniz (1646-1716). So far from being 
an ignoramus, he was one of the greatest logicians of all time, which 
is the more remarkable in that his historical knowledge was rather 
limited. His place in the history of logic is unique. On the one hand 
his achievement constitutes a peak in the treatment of a part of the 
Aristotelian syllogistic, where he introduced many new, or newly 
developed features, such as the completion of the combinatorial 
method, the exact working out of various methods of reduction, 
the method of substitution, the so-called 'Eulerian' diagrams, etc. 
On the other hand he is the founder of mathematical logic. 

The reason why Leibniz is, nevertheless, named in this section, 
and only named, is that his great achievements in the realm of 
mathematical logic are little relevant to the history of problems, 
since they remained for long unpublished and were first discovered 
at the end of the 19th century when the problems he had dealt 
with had already been raised independently. 

Only in one respect does he seem to have exercised a decisive 
influence, in forming the idea of mathematical logic. The pertinent 
passages will be quoted incidentally in the next section. Here we 
limit ourselves to quoting some of his contributions to syllogistic 
theory, and showing some of his diagrams. 

E. COMPREHENSION AND EXTENSION 

The idea underlying the distinction between comprehension and 
extension is a very old one: it is presupposed, for instance, in the 
Isagoge of Porphyry (24.02 ff.); the scholastic doctrine of supposition 

258 



SYLLOGISTIC 

has a counterpart of it in the theory of simple (27.17 IT. ) and personal 
(27.24 fT.) supposition with an elaborate terminology. But the 
expressions comprehension and elendue are first found in the Port 
Royal Logic. Leibniz evidently has the idea, but without an establish- 
ed terminology. 

We first cite an extract from his article De formae logicae com- 
probatione per linearum ductus: 

36.09 Up to now we have assessed the quantities of terms 
in respect of (ex) the individuals. And when it was said: 
'every man is an animal', it was meant (consider alum est) 
that all human individuals form a part of the individuals 
that fall under 'animal' (esse partem individuorum animalis). 
But in respect of (secundum) ideas, the assessment proceeds 
just conversely. For while men are a part of the animals, 
conversely the notion of animal is a part of the notion apply- 
ing to man, since man is a rational animal. 

So Leibniz had a fairly accurate idea of comprehension and 
extension, as well as of their inter-relationship. Now we come to the 
Port Royal Logic: 

36.10 Now in these universal ideas (idees) there are two 
things (choses) which it is important to keep quite distinct: 
comprehension and extension. 

I call the comprehension of an idea the attributes which 
it contains and which cannot be taken away from it without 
destroying it; thus the comprehension of the idea of triangle 
includes extension, figure, three lines, three angles, the 
equality of these three angles to two right-angles, etc. 

I call the extension of an idea the subjects to which it applies, 
which are also called the inferiors of a universal term, that 
being called superior to them. Thus the idea of triangle in 
general extends to all different kinds of triangle. 

F. THE FOURTH FIGURE AND SUBALTERN MOODS 

Already in his youthful work De arte combinatoria Leibniz resumed 
the thought of Albalag (32.25 ff.), without being acquainted with it, 
and proved that there is a fourth figure of assertoric syllogism 
(36.11). Later, he gave a complete and correct table of the twenty- 
four syllogistic moods, in which he deduced the moods of the second 
and third figures from those of the first, using the first reduction 
procedure of Aristotle (§ 14, D). We give a table which reproduces 
the deduction of the second and third figure moods, this time in the 
original language. ' Regressus' means contraposition. 

259 



TRANSITIONAL PERIOD 



36.12 



Barbara primae 


A CD 


ABC 


ABD 


Barbara primae 


ACD 


ABC 


ABD 


Regressus 


ACD 




OBD 


Regr. 




ABC 


OBD 


Ergo 




OBC 




Ergo 


OCD 






Hinc Baroco 








Hinc Bocardo 








secundae 


ACD 


OBD 


OBC 


tertiae 


OBD 


ABC 


OCD 


Celarent primae 


ECD 


ABC 


EBD 


Celarent primae 


ECD 


ABC 


EBD 


Regr. 


ECD 




IBD 


Regr. 




ABC 


IBD 


Ergo 




OBC 




Ergo 


ICD 






Hinc Festino 








Hinc Disamis 








secundae 


ECD 


IBD 


OBC 


tertiae 


IBD 


ABC 


ICD 


Darii primae 


ACD 


IBC 


IBD 


Darii primae 


ACD 


IBC 


IBD 


Regr. 


ACD 




EBD 


Regr. 




IBC 


EBD 


Ergo 




EBC 




Ergo 


OCD 






Hinc Camestres 








Hinc Ferison 








secundae 


ACD 


EBD 


EBC 


tertiae 


EBD 


IBC 


OCD 


Ferio primae 


ECD 


IBC 


OBD 


Ferio primae 


ECD 


IBC 


OBD 


Regr. 


ECD 




ABD 


Regr. 




IBC 


ABD 


Ergo 




EBC 




Ergo 


ICD 






Hinc Cesare 








Hinc Datisi 








secundae 


ECD 


ABD 


EBC 


tertiae 


ABD 


IBC 


ICD 


Barbari primae 


ACD 


ABC 


IBD 


Barbari primae 


ACD 


ABC 


IBD 


Regr. 


ACD 




EBD 


Regr. 




ABC 


EBD 


Ergo 




OBC 




Ergo 


OCD 






Hinc Camestres 








Hinc Felapton 








secundae 


ACD 


EBD 


OBC 


tertiae 


EBD 


ABC 


OCD 


Celaro primae 


ECD 


ABC 


OBD 


Celaro primae 


ECD 


ABC 


OBD 


Regr. 


ECD 




ABD 


Regr. 




ABC 


ABD 


Ergo 




OBC 




Ergo 


ICD 






Hinc Cesaro 








Hinc Darapti 








secundae 


ECD 


ABD 


OBC 


tertiae 


ABD 


ABC 


ICD 



(cf. 13.21, 13.22. 



G. SYLLOGISTIC DIAGRAMS 



The idea of representing class relations and syllogistic moods by 
geometrical figures was familiar to the ancient commentator 
(24.34) ; how far it was current among the Scholastics is not yet 
known. The use of circles is usually ascribed to L. Euler (1701-83) 
(cf. his Lettres a une princesse d'Allemagne, 1768), while that of 
straight lines is associated with the name of Lambert. But the 
former are to be found earlier in J. C. Sturm (1661) (36.13), and 
the latter in Alstedius (1614) (36.14). Schroder (36.15) notes that 
L. Vives was using angles and triangles in 1555.* Leibniz's use 



For the foregoing and some further details vid. A. Menne (36.16) 



260 



p ■ 



n 



&2^ 



I 



'■'7»l <>>***. Me-- ♦;■ 



9~*$ifi* -{ d3 -4 I j 

&»^ r -U-LJ 

• L 1>- V- •> ... ,*C 



LI' a 






£j 



1 ~3i- 









,(M 



?>/*# 




L 



€ 



U3S2> 



o 



Syllogistic diagrams bv Leilmiz 3G. 



36.18 



VENN-DIAGHAMS 

Y Z 






•261 



QUANTIFICATION 

of circles and other diagrammatic methods remained unpublished 
till 1903. We reproduce a page of his MS (36.17) which contains both 
circular and rectilinear diagrams. 

Such methods of presentation were much considered and further 
developed from the time of Euler onwards. J. Venn (1860) intro- 
duced ellipses for his investigations of the relations between more 
than three classes, and marked with a star every region representing 
a non-void class. Three of his diagrams are reproduced in (36.18) 
and a systematic development is considered by W. E. Hocking 
(36.19). 

A different kind of diagram, mnemonic rather than expository 
of probative, due to Johnson, may be added here. 

36.20 The attached diagram, taking the place of the mnemo- 
nic verses, indicates which moods are valid, and which are 
common to different figures. The squares are so arranged that 
the rules for the first, second and third figures also show the 
compartments into which each mood is to be placed, according 
as its major, minor or conclusion is universal or particular, 
affirmative or negative. The valid moods of the fourth figure 
occupy the central horizontal line. 




In the figure, the superscripts V and V indicate the propositions 
that may be weakened or strengthened by subalternation. 



H. QUANTIFICATION OF THE PREDICATE 

While all points so far referred to fall within the general scheme 
of Aristotelian logic, Bentham's doctrine of the quantification of 
the predicate, usually ascribed to Hamilton, is directly opposed to 
Aristotle's teaching (12.03). At the same time, as can be seen from 
the texts, it is a development of the scholastic doctrine of exponibles. 
It has this historical importance, that it shows the kind of problem 
being considered by logicians at the time of Boole, and in some 
degree throws light on the origin of Boole's calculus. 

262 



TRANSITIONAL PERIOD 

We give two texts from G. Bentham (1827) first, then one from 
Hamilton (1860): 

36.21 In the case where both terms of a proposition are 
collective entities, identity and diversity may have place: 

1. Between any individual referred to by one term, and any 
individual referred to by the other. Ex. : The identity between 
equiangular and equilateral triangles. 

2. Between any individual referred to by one term, and 
any one of a part only of the individuals referred to by the 
other. Ex. : The identity between men and animals. 

3. Between any one of a pari only of the individuals referred 
to by one, and any one of a pari only of the individuals referred 
to by the other term. Ex. : The identity between quadrupeds 
and swimming animals. 

36.22 Simple propositions, considered in regard to the above 
relations, may therefore be either affirmative or negative; 
and each term may be either universal or partial. These 
propositions are therefore reducible to the eight following 
forms, in which, in order to abstract every idea not connected 
with the substance of each species, I have expressed the two 
terms by the letters X and Y, their identity by the mathe- 
matical sign =, diversity by the sign || , universality by the 
words in toto, and partiality by the words ex parte; or, for 
the sake of still further brevity, by prefixing the letters I and 
p, as signs of universality and partiality. These forms are, 



1. 


X in toto = 


Y ex parte 


or IX 


= pY 


2. 


X in toto 


Y ex parte 


or IX 


II PY 


3. 


X in toto = 


Y in toto 


or IX 


= IY 


4. 


X in toto 


Y in toto 


or IX 


1 IF 


5. 


X ex parte = 


Y ex parte 


or pX 


= PY 


6. 


X ex parte 


Y ex parte 


or pX 


1 PY 


7. 


X ex parte = 


Y in toto 


or pX 


= tY 


8. 


X ex parte \\ 


Y in toto 


or pX 


II tY 



Hamilton writes: 

36.23 The second cardinal error of the logicians is the not 
considering that the predicate has always a quantity in 
thought, as much as the subject; although this quantity be 
frequently not explicitly enounced, as unnecessary in the 
common employment of language ; for the determining notion 
or predicate being always thought as at least adequate to, or 

263 



QUANTIFICATION 

coextensive with, the subject or determined notion, it is 
seldom necessary to express this, and language tends ever 
to elide what may safely be omitted. But this necessity 
recurs, the moment that, by conversion, the predicate becomes 
the subject of the proposition; and to omit its formal state- 
ment is to degrade Logic from the science of the necessities 
of thought, to an idle subsidiary of the ambiguities of speech. 
An unbiased consideration of the subject will, I am confident, 
convince you that this view is correct. 

1°, That the predicate is as extensive as the subject is 
easily shown. Take the proposition, - 'All animal is man', or, 
'All animals are men'. This we are conscious is absurd. . . . 
We feel it to be equally absurd as if we said, - 'All man is all 
animal', or ,'A11 men are all animals'. Here we are aware that 
the subject and predicate cannot be made coextensive. If we 
would get rid of the absurdity, we must bring the two notions 
into coextension, by restricting the wider. If we say - 'Man 
is animal', [Homo est animal], we think, though we do not 
overtly enounce it, 'All man is animal'. And what do we mean 
here by animal? We do not think, all, but some, animal. And 
then we can make this indifferently either subject or predi- 
cate. We can think, - we can say, 'Some animal is man', 
that is, some or all man; and, e converso, 'Man (some or all) is 
animal', viz. some animal. . . . 

2°, But, in fact, ordinary language quantifies the predicate 
so often as this determination becomes of the smallest import. 
This it does either directly, by adding all, some, or their 
equivalent predesignations to the predicate ; or it accomplishes 
the same end indirectly, in an exceptive or limitative form. 

Hamilton then proceeds to repeat, in dependence on the works 
of various 17th and 18th century logicians, the scholastic doctrine 
of the exponibilia (§ 34, C). 



264 



PART V 

The mathematical variety of Logic 



MATHEMATICAL LOGIC 

I. General Foundations 
§37. INTRODUCTION TO MATHEMATICAL LOGIC 

A. CHARACTERISTICS 

The development of the mathematical variety of logic is not yet 
complete, and discussions still go on about its characteristic scope 
and even about its name. It was simultaneously called 'mathemati- 
cal logic', 'symbolic logic' and 'logistic' by L. Couturat, Itelson and 
Lalande in 1901, and is sometimes simply called 'theoretical logic'. 
Even apart from the philosophical discussions as to whether or 
how far it is distinct from mathematics, there is no unanimity about 
the specific characteristics which distinguish it from other forms of 
logic. 

However, there exists a class of writings which are generally 
recognized as pertaining to 'mathematical logic' ('logistic', 'symbolic 
logic' etc.). Analysis of their contents shows that they are predomi- 
nantly distinguished from all other varieties of logic by two inter- 
dependent characteristics. 

(1) First, a calculus, i.e. a formalistic method, is always in evi- 
dence, consisting essentially in the fact that the rules of operation 
refer to the shape and not the sense of the symbols, just as in mathe- 
matics. Of course formalism had already been employed at times 
in other varieties of logic, in Scholasticism especially, but it is now 
erected into a general principle of logical method. 

(2) Connected with that is a deeper and more revolutionary 
innovation. All the other varieties of logic known to us make use 
of an abslractiue method ; the logical theorems are gained by abstrac- 
tion from ordinary language. Mathematical logicians proceed in 
just the opposite way, first constructing purely formal systems, and 
later looking for an interpretation in every-day speech. This process 
is not indeed always quite purely applied; and it would not be 
impossible to find something corresponding to it elsewhere. But at 
least since Boole, the principle of such construction is consciously 
and openly laid down, and holds sway throughout the realm of 
mathematical logic. 

Those are the essential features of mathematical logic. Two more 
should be added: 

(3) The laws are formulated in an artificial language, and consist 
of symbols which resemble those of mathematics (in the narrower 
sense). The new feature here is that even the constants are expressed 
in artificial symbols; variables, as we have seen, have been in use 
since the time of Aristotle. 

(4) Finally, until about 1930 mathematical logic formulated its 

266 



CHRONOLOGY 

theorems in an object language, in this unlike the Scholastics, but 
in conformity with the ancients. That this is no essential feature 
is shown by more recent developments and the spread of metalogical 
formulation. But till 1930 the use of the object language is charac- 
teristic. 

It may be further remarked that it can be said of mathematical 
logic, what was finally said about scholastic, that it is very rich 
and very formalistic. In wealth of formulae indeed, it seems to 
exceed all other forms of logic. It is also purely formal, being sharply 
distinguished from the decadent 'classical' logic by its avoidance of 
psychological, epistemological and metaphysical questions. 

B. CHRONOLOGICAL SEQUENCE 

G. W. Leibniz generally ranks as the original mathematical 
logician, but if he cannot count as the founder of mathematical 
logic it is because his logical works were for the most part published 
long after his death (the essentials by L. Couturat in 1901). However, 
he had some successors, the most important of whom were the 
brothers Bernoulli (1685), G. Plouquet (1763, 1766), J. H. Lambert 
(1765, 1782), G. J. von Holland (1764), G. F. Castillon (1803) and 
J. D. Gergonne (1816/17).* But no school arose. 

One who did found a school, and who stands at the beginning of 
the continuous development of mathematical logic, is George 
Boole, whose first pioneer work, The Mathematical Analysis of 
Logic, appeared in 1847. In the same year Augustus de Morgan 
published his Formal Logic. Boole's ideas were taken further in 
different directions by R. L. Ellis (1863), W. S. Jevons (1864), 
R. Grassmann (1872), J. Venn (1880, 1881), Hugh McColl (1877/78), 
finally and chiefly by E. Schroder (1877, 1891-95). 

Contemporaneous with the last-named are the works of a new 
group of mathematical logicians whose chief representatives are 
C. S. Peirce (1867, 1870), Gottlob Frege (1879), and G. Peano 
(1888). Of these three important thinkers only Peano founded a 
considerable school; Peirce and Frege went practically unnoticed. 
It was Bertrand Russell (1903) who discovered the thought of 
Frege and together with A. N. Whitehead combined it with his 
own discoveries in Principia Mathematica (1910-13), in which the 
symbolism of Peano was used. 

D. Hilbert (1904) and L. E. J. Brouwer (1907, 1908) were active 
before the appearance of the Principia. J. Lukasiewicz published 
his first work in this field in 1910, St. Lesniewski in 1911. They 
were followed by A. Tarski (1921), R. Carnap (1927), A. Heyting 
(1929) and K. Godel (1930). 

* Figures in parentheses give the year of publication of the main work, 
then of the first subsequent important one. 

267 



MATHEMATICAL LOGIC 



These are only a few of the great number of mathematical logi- 
cians, which by now is beyond count. 



C. FREGE 

Among all these logicians, Gottlob Frege holds a unique place. 
His Begriffsschrift can only be compared with one other work in 
the whole history of logic, the Prior Analytics of Aristotle. The two 
cannot quite be put on a level, for Aristotle was the very founder of 
logic, while Frege could as a result only develop it. But there is a 
great likeness between these two gifted works. The Begriffsschrift, 
like the Prior Analytics, contains a long series of quite new insights, 
e.g. Frege formulates for the first time the sharp distinction between 
variables and constants, the concepts of logical function, of a many- 
place function, of the quantifier; he has a notably more accurate 
understanding of the Aristotelian theory of an axiomatic system, 
distinguishes clearly between laws and rules, and introduces an 
equally sharp distinction between language and meta-language, 
though without using these terms; he is the author of the theory of 
description; without having discovered, indeed, the notion of a 
value, he is the first to have elaborated it systematically. And that 
is far from being all. 

At the same time, and just like Aristotle, he presents nearly all 
these new ideas and intuitions in an exemplarily clear and systematic 
way. Already in the Begriffsschrift we have a long series of mathe- 
matico-logical theorems derived from a few axioms 'without 
interruption' (luckenlos), as Frege says, for the first time in history. 
Various other mathematical logicians at the same time, or even 
earlier, expounded similar ideas and theories, but none of them had 
the gift of presenting all at once so many, often quite original, 
innovations in so perfect a form. 

It is a remarkable fact that this logician of them all had to wait 
twenty years before he was at all noticed, and another twenty before 
his full strictness of procedure was resumed by Lukasiewicz. In this 
last respect, everything published between 1879 and 1921 fell below 
the standard of Frege, and it is seldom attained even today. The 
fate of Frege's work was in part determined by his symbolism. 
It is not true that it is particularly difficult to read, as the reader 
can assure himself from the examples given below; but it is certainly 
too original, and contrary to the age-old habits of mankind, to be 
acceptable. 

All that we have said does not mean that Frege is the only great 
logician of the period now under consideration. We also have to 
recognize as important the basic intuitions of Boole, and many 
discoveries of Peirce and Peano, to name only these three. The very 
fact that Frege was a contemporary of Peirce and Peano forbids 

268 



PERIODS 

one to treat him as another Aristotle. But of all mathematical 
logicians he is undoubtedly the most important. 

D. PERIODS 

The history of mathematical logic can be divided into four periods. 

1. Prehistory: from Leibniz to 1847. In this period the notion of 
mathematical logic arose, and many points of detail were formulated, 
especially by Leibniz. But there was no school at this time, and 
the continuous development had not yet begun. There were, rather, 
isolated efforts which went unnoticed. 

2. The Boolean period, from Boole's Analysis to Schroder's Vor- 
lesungen (vol. I, 1895). During this period there is a continuous 
development of the first form of mathematical logic. This form is prin- 
cipally distinguished from later ones in that its practitioners did not 
make the methods of mathematics their object of study, but con- 
tented themselves with simply applying them to logic. 

3. The period of Frege, from his Begriffsschrift (1879) to the 
Principia Mathematica of Whitehead and Russell (1910-13). Frege, 
and contemporaneously Peirbe and Peano, set a new goal, to find 
foundations for mathematics. A series of important logical ideas 
and methods were developed. The period reaches its peak with the 
Principia which both closes the preceding line of development and 
is the starting point of a new one, its fruitfulness being due in the 
first place to a thorough consideration and solution of the problem 
of the antinomies which had been a burning question since the end 
of the 19th century and had not previously found a solution in the 
new period. 

4. The most recent period : since the Principia, and still in progress. 
This period can be sub-divided: the years from 1910 to 1930 are 
distinguished by the rise of metalogic, finitist in Hilbert, not so 
in Lowenheim and Skolem; after about 1930 metalogic is systema- 
tized in a formalistic way, and we have Tarski's methodology, 
Carnap's syntax, and the semantics of Godel and Tarski in which 
logic and metalogic are combined. The 'natural' logics of Gentzen 
and Jaskowski (1934) also belong here. 

So we can say that the advance of metalogic is distinctive of the 
time since 1910, though new logical systems (in the object language) 
continue to appear: that of Lewis (1918), the many-valued systems of 
Post and Lukasiewicz (1920-21), the intuitionistic logic of Heyting 
(1930). Finally the very original systems of combinatorial logic 
by Schonfinkel (1924), Curry (1930), Kleene (1934), Rosser (1935) 
and Church (1936-41). 

This fourth period will be touched on only very lightly, in some 
of its problems. 

The following table gives an easy view of the temporal sequence 

269 



MATHEMATICAL LOGIC 

of the logicians we have named. But it is to be noticed (1) that 
temporal succession does not always reflect actual influence; this 
will be discussed more in detail in the various chapters. (2) the subject 
developed so fast after 1870 that dates of births and deaths are 
little to the purpose ; we have preferred to give those of publication 
of the chief logical works. 

G. W. v. Leibniz 



A. De Morgan 1847 



C. S. Peirce 1867-1870 



G. Frege 1879 
G. Peano 1888 



G. Boole 1847 

R. C. Ellis 1863 

W. S. Jevons 1864 

R. Grassmann 1872 

H. McColl 1877/78 

E. Schroder 1877 



D. Hilbert 1904 
L. Brouwer 1907/8 



B. Russell 1903 
Principia 1910-1913 



J. Lukasiewicz 1910 
St. Lesniewski 1911 
A. Tarski 1921 
R. Carnap 1927 
A. Heyting 1929 
K. Godel 1930 



E. STATE OF RESEARCH 



Mathematical logic is the best known form of logic, since many 
of its basic works, especially the Principia, so far from being past 
history are still in current use. Then again there have already been 
a number of historical studies of the period. Among these are the 
work of B. Jourdain (37.01), the historical sections of the works of 
C. I. Lewis (37.02) and J. Jorgensen (37.03). The treatise of H. Hermes 
and H. Scholz (37.04) is remarkably rich in historical information. 
Since 1936 we have had as unique tools of research, a biblio- 
graphy of mathematical logic from Leibniz to 1935, and the Journal 
of Symbolic Logic containing a current bibliography and excellent 
indexes. Both are as good as bibliograj hy can be, under the editor- 
ship of A. Church who sees to them with exemplary punctuality and 
regularity. Among other contributions to the history of this period 
the numerous papers of R. Feys should be mentioned. 



270 



METHOD 

But still we do not know all about the period. L. Couturat's 
thorough and serious monograph on Leibniz needs completing on 
many points in the light of more recent systematic and historical 
research; there are also various other treatises on Leibniz's logic. 
Boole, too, has been fairly thoroughly investigated in recent years. 
But as yet there is no detailed treatment of Leibniz's successors, no 
monograph on Peirce, above all no thorough work on Frege's logic, 
without mentioning other less important logicians. 

F. METHOD 

For the reason stated in the introduction, we have tried to present 
the essential range of problems discussed in mathematical logic by 
means of texts containing little or no artificial symbols. This has 
proved feasible by and large, but not without exception ; in particular, 
at least the basic methods have to be explained in terms of the 
contemporary symbolism, e.g. of Frege or the Principia. Then again, 
we have given the most important theorems in the various fields 
in symbolic formulation, in order to facilitate comparison with 
similar theorems developed in other periods. 

The question of what time-limit to put is very difficult, and the 
various periods within the main one dovetail into each other in such 
a way as to make the drawing of sharp boundaries impossible. We 
have finally decided to close the exposition with the Principia, 
touching lightly on a few later developments which are either 
closely connected with matters discussed before 1910, or of special 
interest on their own account. That section (§ 49) is accordingly in 
the nature of an appendix. 

The reader will only be able to appreciate the textual fragments that 
follow if, first, he is well acquainted with the fundamental concepts 
of contemporary mathematical logic (cf. § 5, B); second, he is able to 
abstract from the philosophy (ontology, epistemology, psychology 
etc.) of the various logicians. For never before have formal logicians 
been so divided by mutually opposed philosophies as here. We need 
only instance Frege's outspoken Platonism, and Boole's nominalism 
and even psychologism. But they have all developed essentially the 
same formal logic. 

That is not to say that the individual philosophic views have been 
entirely without influence on the form of this or that system. But 
such influence has been much slighter than an unbiased observer 
might at first suppose. That the systems present such different 
appearances is due mainly to differences of immediate purpose 
(one may compare Boole with Frege, or Peano with Lukasiewicz), 
and to differences in the degree of exactness which are more marked 
here than in any other period. 



271 



§38. METHODS OF MATHEMATICAL LOGIC 

Two essentially distinct methodological ideas seem to underlie 
mathematical logic. On the one hand it is a logic that uses a calculus. 
This was developed in connection with mathematics, which at first 
was considered as the ideal to which logic should approach. On the 
other hand mathematical logic is distinguished by the idea of exact 
proof. In this respect it is no hanger-on of mathematics, and this is 
not its model; it is rather the aim of logic to investigate the founda- 
tions and conduct of mathematics by means of more exact methods 
than have been customary among 'pure' mathematicians, and to 
offer to mathematics the ideal of strict proof. 

In both respects the name 'mathematical logic' is justified, 
though for opposite reasons; first, because the new logic is a result of 
mathematics, then because it seeks to provide a basis for that 
science. But it would be a misunderstanding to conclude that mathe- 
matical logicians want to confine themselves to the consideration of 
quantities; their aim from the start has rather been to construct a 
quite general logic. 

In what follows we illustrate both aspects with a series of texts 
which resume the development of mathematical logic. 



A. LOGICAL CALCULUS 
1. Lull 

The idea of a mechanical process to facilitate inference is already 
present in the combinatorial arguments of the ancient Commentators, 
the Arabs and the Scholastics. We have given one example above 
(32.34), but of course it was only a matter of determining correct 
syllogistic moods. Raymond Lull (1235-1315) is the first to lay claim 
to a quite general mechanical procedure. It appears from the work 
of this remarkable man that he believed himself to have found a 
method which permits one to draw every kind of conclusion by 
means of a system of concentric, circular sheets or rings, of various 
sizes and mutually adjustable, with letters inscribed on their rims. 
Unfortunately Lull does not express the main ideas of this procedure 
at all clearly. However, it will be well to give at least a few passages 
from his Ars Magna, since his doctrine is not only one of the greatest 
curiosities in the history of logic, but also had some influence on 
Leibniz. 

38.01 The understanding longs and strives for a universal 
science of all sciences, with universal principles in which the 
principle of the other, more special sciences would be implicit 
and contained as is the particular in the universal. . . . 

272 



CALCULUS 

38.02 This art is divided into thirteen parts, viz. the 
alphabet, the figures, definitions, rules, tables . . . (etc.). 

The alphabet of this art is the following: 

B signifies goodness, difference, whether, God, justice, 
avarice. 

C signifies quantity, conformity, what, angel, prudence, 
throat. 

We spare the reader the further enumeration of this alphabet. 
But we print a picture of the 'first figure' and here is part of the 
accompanying commentary: 

38.03 There are four figures, as appears from this page. 
The first figure is signified by A; and it is circular, subdivided 
into nine compartments. In the first compartment is B, in 
the second C, etc. And it is said to be cruciform, in that the 
subject is turned into the predicate and conversely, as when 
one says: great goodness, good greatness ; eternal greatness, 
great eternity; God the good (Deus bonus), the good God 
(bonus Deus), and correspondingly for other (terms). By 
means of rotations of this kind the practitioner (artista) can 
see what is converted and what is not converted, such as 
'God is good' and the like, which can be converted. But God 
and angel will not be converted, nor goodness and angel, 
nor its goodness and (its) greatness, and so on with the other 
terms. 

This text is far from clear, and its consequences no clearer; it has, 
moreover, little relevance for genuine logic. But the mere idea of such 
a mechanical process was a fascinating one for many people in the 
16th and 17th centuries. 

2. Hobbes 

Lull's ideas are to be found expressed in an extreme form three 
hundred years later, by Thomas Hobbes (1655). He made no attempt 
to carry them out, for like most modern philosophers Hobbes was no 
logician. 

38.04 By ratiocination I mean computation. Now to 
compute, is either to collect the sum of many things that 
are added together, or to know what remains when one thing 
is taken out of another. Batiocination, therefore, is the same 
with addition and substruction (sic); and if any man add 
multiplication and division, I will not be against it, seeing 
multiplication is nothing but addition of equals one to 

273 



MATHEMATICAL LOGIC 

another, and division nothing but a substraction of equals 
one from another, as often as is possible. So that all ratio- 
cination is comprehended in these two operations of the mind, 
addition and substraction. 

This is, to be sure, rather the jeu d' esprit of a dilettante than a 
theory of mathematical logic; no inference can be interpreted in this 
way, and Hobbes never once tried to do it. The passage shows the 
mark of his extreme verbalism, inference being a mere accumulation 
of words. However, this text is historically important as having 
exercised some influence on Leibniz, and it is also characteristic 
of the mathematicism which largely dominated the new form of 
logic until Jevons. But perhaps no logician was so badly infected 
with it as Hobbes. 

3. Leibniz 

Leibniz had read Lull (38.05) and cites Hobbes too (38.06). But he 
has much more to offer than either ol them. Like Lull, he is con- 
cerned with a universal basis for all sciences; like Lull again, his 
basic philosophy leads him to think of a purely combinatorial 
method. But this is now to take the form of a calculus, such as is 
employed in mathematics ; logic is to be thought of as a generalized 
mathematics. Leibniz's most characteristic texts on this point are 
the following: 

38.07 As I was keenly occupied with this study, I happened 
unexpectedly upon this remarkable idea, that an alphabet 
of human thought could be devised, and that everything 
could be discovered (inveniri) and distinguished (dijudicari) 
by the combination of the letters of this alphabet and by the 
analysis of the resulting words. 

38.08 The true method should afford us a filum Ariadnes, 
i.e. a sensibly perceptible and concrete means to guide the 
mind, like the lines drawn in geometry and the forms of the 
operations which are prescribed to learners in arithmetic. 
Without this our mind could traverse no path without going 
astray. 

38.09 To discover and prove truths, the analysis of ideas 
is necessary, . . . which corresponds to the analysis of (written) 
characters. . . . Hence we can make the analysis of ideas 
sensibly perceptible and conduct it as with a mechanical 
thread; since the analysis of the characters is something 
sensibly perceptible. 

274 




The "first figure" of Lull's "Ars Magna". Cf. 38.03 



CALCULUS 

38.10 A characteristic of reason, by means of which truths 
would become available to reason by some method of calcula- 
tion, as in arithmetic and algebra, so in every other domain, 
so long as it submits to the course of deduction. 

38.11 Then, in case of a difference of opinion, no discussion 
between two philosophers will be any longer necessary, as 
(it is not) between two calculators. It will rather be enough 
for them to take pen in hand, set themselves to the abacus, 
and (if it so pleases, at the invitation of a friend) say to one 
another: Calculemus! 

38.12 Ordinary languages, though mostly helpful for the 
inferences of thought, are yet subject to countless ambiguities 
and cannot do the task of a calculus, which is to expose 
mistakes in inference owing to the forms and structures of 
words, as solecisms and barbarisms. This remarkable advan- 
tage is afforded up to date only by the symbols (notae) of 
arithmeticians and algebraists, for whom inference consists 
only in the use of characters, and a mistake in thought and 
in the calculus is identical. 

38.13 Hence it seems that algebra and the mathesis uni- 
versalis ought not to be confused with one another. If indeed 
mathesis was to deal only with quantity, or with equals and 
unequals, or with mathematical ratio and proportion, there 
would be nothing to prevent algebra (which considers quan- 
tity in general) from being considered as their common part. 
But mathesis seems to underlie whatever the power of imagina- 
tion underlies, insofar as that is accurately conceived, and so 
it pertains to it to treat not only of quantity but also of the 
arrangement (dispositio) of things. Thus mathesis universalis, 
if I am not mistaken, has two parts, the ars combinatoria 
concerned with the variety of things and their forms or 
qualities in general insofar as they are subject to exact 
inference, and the equal and the unequal; then logistic or 
algebra, which is about quantity in general. 

There are here two different, though connected ideas: that of an 
'alphabet of thought' and that of mathesis universalis. According 
to the first, one is to assign a symbol to every simple idea and solve 
all problems by combinations of these symbols. This is very con- 
sonant with Leibniz's philosophy, in particular with his doctrine of 
the strictly analytic character of all necessary propositions and of 
inference as a combining of elements. This philosophical view, 
questionable in itself, was yet fruitful for logic in that it led Leibniz 

275 



MATHEMATICAL LOGIC 

to the notion of an artificial language (38.12) which, by contrast to 
ordinary languages, would be free from ambiguities. Therein Leibniz 
is the founder of symbolic logic as such, i.e. of the use of artificial 
symbols even for logical constants (and not only for variables as in 
all earlier forms of logic). 

The other idea is that of mathesis universalis (38.10), i.e. of the 
application of calculation to all inferences, not only to those that 
are mathematical in the narrower sense. Leibniz does not advocate 
any mathematicism such as that of Hobbes: malhesis universalis is 
sharply distinguished from ordinary algebra (here strangely called 
'logistic') and set in contrast to it (38.13). It is only the method that 
is to be applied to logic, and this is not any 'addition and subtrac- 
tion' as with Hobbes, but simply formal operating with symbols. Of 
course the idea of a strictly formalistic logic, of constructing some 
meaningless system which is only interpreted later, such as we find in 
Boole, is not yet present. The calculus is to be a filum Ariadnes to 
assist the mind. The process envisaged is therefore basically the 
same as in the earlier logical tradition; formal laws are abstracted 
from meaningful sentences. But the principle of a formal process, i.e. 
of calculation, is here clearly expressed for the first time, so far as we 
know. In this, Leibniz is the founder of mathematical logic. 

4. Lambert 

Some further development of Leibniz's ideas is to be found in 
Lambert (1728-1777): 

38.14 Let us see, then, how a more universal idea can be 
abstracted from the arithmetical and algebraic calculi. First, 
the idea of quantity must be got rid of, as being too special. 
You may put in its place qualities, affections, thinys, truths, 
ideas and whatever can be discussed, combined, connected, 
separated and chanyed into ever new forms; all and each of these 
substitutions can be made in accordance with the nature of 
things. For each of these operations and changes, with due 
differences allowed for, are applied to quantities. 

Further, for the ideas of equality, equation, ratio, relation, 
proportion, proyression, etc. which occur in arithmetic, more 
universal ones are to be substituted. So that in place of 
equality it will be convenient to introduce identity, in place of 
equation identification, if this word is taken in its active sense, 
in place of proportion analoyy. And if the words relation, 
proyression be retained, their meaning is to be extended, as 
ordinary usage suggests, so that they can be thought of as 
relations or progressions between the thinys, qualities, 

276 



CALCULUS 

affections, ideas or truths to which the calculus is to be suited. 
And this is to be chiefly noted, that those relations contribute 
no little to determining the form of the calculus, and that all 
those operations which the object of the calculus admits rest chiefly 
on them. 

5. Gergonne 

Gergonne (1816/17) comes much closer to the idea of formalism: 

38.15 It is constantly being said that reasoning must only 
be about objects of which one has a perfectly clear idea, yet 
often nothing is more false. One reasons, in practice, with 
words, just as one calculates with letters in algebra; and in 
the same way that an algebraic calculation can be carried out 
exactly without one having the slightest idea about the sig- 
nification of the symbols on which one is operating, in the 
same way it is possible to follow a course of reasoning without 
any knowledge of the signification of the terms in which it is 
expressed, or without adverting to it if one knows it. . . . It is 
evident, for example, that one does not have to know the 
nature of the terms of a proposition in order to deduce its 
converse or subaltern when it admits of such. Doubtless one 
cannot dispense with a good knowledge of notions which are to 
be the immediate matter of judgment; but that is quite 
unnecessary for concluding to a judgment from a number of 
others already known to be correct. 

This text is not altogether clear; Gergonne seems to equate the 
(Aristotelian) use of variables with formalism. But we can see the 
idea of formalism becoming clearer. 

Gergonne also gave expression to an idea which is not without 
relevance to the symbolism of mathematical logic: 

38.16 There is no known language in which a proposition 
exactly and exclusively expresses in which of our five relations 
both its component terms stand; such a language would have 
five kinds of proposition and its dialectic would be quite 
different from that of our languages. 

He is referring there to five relationships between the extensions 
of two terms (or classes) which will be spoken of later (40.12). * 

* We learned of this passage from a work of J. A. Faris. 

277 



MATHEMATICAL LOGIC 
6. Boole 

We can find a clear idea of formalism, developed in an exemplary 
way, in the introduction to George Boole's epoch-making The 
Mathematical Analysis of Logic (1847), in this superior to much later 
works, e.g. the Principia. 

38.17 They who are acquainted with the present state of 
the theory of Symbolical Algebra, are aware, that the validity 
of the processes of analysis does not depend upon the inter- 
pretation of the symbols which are employed, but solely upon 
the laws of their combination. Every system of interpretation 
which does not affect the truth of the relations supposed, is 
equally admissible, and it is thus that the same process may, 
under one scheme of interpretation, represent the solution of a 
question on the properties of numbers, under another, that 
of a geometrical problem, and under a third, that of a problem 
of dynamics or optics. This principle is indeed of fundamental 
importance; and it may with safety be affirmed, that the 
recent advances of pure analysis have been much assisted by 
the influence which it has exerted in directing the current of 
investigation. 

But the full recognition of the consequences of this impor- 
tant doctrine has been, in some measure, retarded by acciden- 
tal circumstances. It has happened in every known form of 
analysis, that the elements to be determined have been con- 
ceived as measurable by comparison with some fixed standard. 
The predominant idea has been that of magnitude, or more 
strictly, of numerical ratio. The expression of magnitude, or 
of operations upon magnitude, has been the express object 
for which the symbols of Analysis have been invented, and 
for which their laws have been investigated. Thus the abstrac- 
tions of the modern Analysis, not less than the ostensive 
diagrams of the ancient Geometry, have encouraged the 
notion, that Mathematics are essentially, as well as actually, 
the Science of Magnitude. 

The consideration of that view which has already been 
stated, as embodying the true principle of the Algebra of 
Symbols, would, however, lead us to infer that this conclusion 
is by no means necessary. If every existing interpretation is 
shewn to involve the idea of magnitude, it is only by induction 
that we can assert that no other interpretation is possible. 
And it may be doubted wither our experience is sufficient to 

278 



CALCULUS 

render such an induction legitimate. The history of pure 
Analysis is, it may be said, too recent to permit us to set 
limits to the extent of its applications. Should we grant to the 
inference a high degree of probability, we might still, and with 
reason, maintain the sufficiency of the definition to which the 
principle already stated would lead us. We might justly 
assign it as the definitive character of a true Calculus, that it is 
a method resting upon the employment of Symbols, whose 
laws of combination are known and general, and whose 
results admit of a consistent interpretation. That to the 
existing forms of Analysis a quantitative interpretation is 
assigned, is the result of the circumstances by which those 
forms were determined, and is not to be construed into a 
universal condition of Analysis. It is upon the foundation of 
this general principle, that I purpose to establish the Calculus of 
Logic, and that I claim for it a place among the acknowledged 
forms of Mathematical Analysis, regardless that in its object 
and in its instruments it must at present stand alone. 
From that Boole draws the explicit conclusion: 

38.18 On the principle of a true classification, we ought no 
longer to associate Logic and Metaphysics, but Logic and 
Mathematics. . . . The mental discipline which is afforded by 
the study of Logic, as an exact science, is in species, the same 
as that afforded by the study of Analysis. 

Leibniz and Lambert had already wanted to apply calculation 
to logic, and had used the idea of non-quantitative calculation. The 
epoch-making feature of Boole's text is the exemplarily clear account 
of the essence of calculation, viz. formalism, a process of which the 
'validity does not depend upon the interpretation of the symbols 
which are employed, but solely upon the laws of their combination'. 
Boole is also aware of the possibility of interpreting the same formal 
system in different ways. This suggests that he did not think of logic 
as an abstraction from actual processes, as all previous logicians had 
done, but as a formal construction for which an interpretation is 
sought only subsequently. That is quite new, and in contrast with 
the whole tradition, Leibniz included. 

7 '. Peirce 

Finally we submit a text from Peirce's review of Schroder's logic 
(1896), which contains one of the best statements of the advantage 
to be looked for in a logical calculus. 

38.19 It is a remarkable historical fact that there is a 
branch of science in which there has never been a prolonged 

279 



MATHEMATICAL LOGIC 

dispute concerning the proper objects of that science. It is 
mathematics. Mistakes in mathematics occur not infrequently, 
and not being detected give rise to false doctrine, which may 
continue a long time. Thus, a mistake in the evaluation of a 
definite integral by Laplace, in his Mecanique celeste, led to an 
erroneous doctrine about the motion of the moon which 
remained undetected for nearly half a century. But after the 
question had once been raised, all dispute was brought to a 
close within a year. . . . 

38.20 Hence, we homely thinkers believe that, considering 
the immense amount of disputation there has always been 
concerning the doctrines of logic, and especially concerning 
those which would otherwise be applicable to settle disputes 
concerning the accuracy of reasonings in metaphysics, the 
safest way is to appeal for our logical principles to the science 
of mathematics, where error can only long go unexploded on 
condition of its not being suspected. . . . 

38.21 Exact logic will be that doctrine of the conditions of 
establishment of stable belief which rests upon perfectly 
undoubted observations and upon mathematical, that is, 
upon diagrammatical, or iconic, thought. We, who are sectaries 
of 'exact' logic, and of 'exact' philosophy, in general, main- 
tain that those who follow such methods will, so far as they 
follow them, escape all error except such as will be speedily 
corrected after it is once suspected. 



B. THEORY OF PROOF 



1. Bolzano 



A noteworthy precursor of modern proof-theory is Bernard 
Bolzano*. 

38.22 If we now state that M, N, 0, . . . are deducible from 
A, B, C, . . . and this in respect of the notions i, /, . . .: we are 
basically saying, according to what has been said in § 155, 
the following: 'All ideal contents which in the place of i, /, . . . 
in the propositions A, B, C, . . . M, N, 0, . . . simultaneously 
verify the propositions A, B, C, . . . has the property of also 
simultaneously verifying the propositions M, TV, 0. . . .' The 



* Professor Hans Hermes drew our attention to this passage. 

The rows of dots after the groups of letters are here part of the text. 



280 



PROOF 

most usual way of giving expression to such propositions is of 
course: '// A, B, C, . . . are true: then also M, TV, 0, . . . are 
true.' But we often also say: 'M, TV, 0, . . . follow, or are Reduc- 
ible, or can be inferred from A, B, C, . . . etc' In respect 
of the notions i, /',... which we consider as the variables in 
these propositions, the same remark is applicable as in No. 1. 
But since according to § 155 No. 20 it is not at all the < •;)-«■ 
with the relation of deducibility, as (it is) with the relations of 
mere compatibility, that a given content of propositions 
A, B, C, .... on the one hand, and M, TV, 0, ... on the other, 
can come into this relationship merely because we determine 
arbitrarily which notions therein are to count as variables: 
it is thus a rather startling statement when we say that certain 
propositions M, N, 0, . . . can be brought into a relationship of 
deducibility with other propositions A, B, C, ... by merely 
taking the notions pertaining to them as variable. But in such 
a judgment we only say that there are certain parts of the 
propositions A, B, C, . . . M, TV, 0, . . . which can be considered 
as variable, with the result that every ideal content which in 
the place of i, /, . . . makes all of A, B, C, . . . true, also makes 
all of M, TV, 0, . . . true. And thus we can easily see from 
§ 137 how such a proposition must be expressed to bring out 
its logically constant parts. 'The notion of some parts of 
A, B, C, . . . M, TV, 0, ... so constituted that every arbitrary 
ideal content which in their place verifies A, B, C, . . . always 
also verifies M, TV, 0, . . . has objectivity'. In ordinary speech 
propositions of this kind are expressed just like the preceding 
ones. It is only from other circumstances, e.g. from the 
context, that one can guess whether the speaker has in his 
mind determinate notions in respect to which the retation of 
deducibility is to be present, or whether he only inlends to 
intimate that there are such notions. Thus, e.g., it is easy 
enough to gather from the following proposition: 'if Caius is a 
man, and all men are mortal, then Caius, too, is mortal', that 
it is here intended to state the deducibility of the proposition : 
Caius is mortal, from the two propositions : Caius is a man, and, 
all men are mortal, in respect of the three notions; Caius, man 
and mortal. This next utterance on the other hand: 'If in all 
men there stirs an irresistible desire for permanence ; if, too, the 
most virtuous must feel unhappy at the thought that he is 
one day to cease; then we are not wrong to expect of God's 
infinite goodness that he will not annihilate us in death' - 

281 



MATHEMATICAL LOGIC 

this would be subject to the reproach of extreme obscurity, 
since its sense is not that the said propositions stand in a 
relationship of deducibility when some of their notions (which 
still have to be ascertained) have been taken as variable. 
By such an utterance it is only intended to state that notions 
are present such as to warrant inference from the truth of the 
antecedent to the truth of the consequent; but it does not as 
yet tell one which these notions properly are. 

2. Frege 

While that text of Bolzano's contains important ideas about the 
concept of deduction or deducibility, the modern development of this 
second aspect of mathematical logic begins with Frege. We take the 
essential texts from the Grundgesetzen der Arilhmeiik (1893); but it 
can easily be shown that most of what is said in them was already 
known to this great logician by 1879. 

38.23 In my Grundlagen der Arithmetik I have tried to 
make it plausible that arithmetic is a branch of logic and does 
not need to take its grounds of proof either from experience or 
intuition. This will now be confirmed in the present book, by 
the fact that the simplest laws of numbers can be deduced by 
logical means alone. But at the same time this shows that 
considerably higher demands must be made on the process of 
proof than is usual in arithmetic. A region of some ways of 
inference and deduction must be previously delimited, and no 
step may be made which is not in accordance with one of 
these. In the passage, therefore, to a new judgment, one must 
not be satisfied with the fact that it is evidently correct, as 
mathematicians nearly always have been up to now, but one 
must analyze it into its simple logical steps, which are often by 
no means a few. No presupposition may remain unremarked ; 
every axiom which is needed must be discovered. It is just the 
tacit presuppositions, that are made without clear conscious- 
ness, which obscure understanding of the epistemological cha- 
racter of a law. 

38.24 The ideal of a strict scientific method in mathematics, 
such as I have here tried to realize, and which could well be 
called after Euclid, I might describe thus. It cannot indeed 
be required that everything should be proved, since that is 
impossible; but one can see to it that all propositions which 
are used without being proved, are expressly stated as such, 

282 



PROOF 

so that it is clearly known on what the whole structure rests. 
The effort, then, must be to reduce the number of these 
primitive laws as far as possible, by proving everything which 
can be proved. Further, and here I go beyond Euclid, I 
require that all methods of inference and deduction which are 
to be applied, shall be previously presented. Otherwise it is 
impossible to ensure with certainty that the first requirement 
is fulfilled. I think that I have now attained this ideal in 
essentials. Stricter requirements could only be made in a few 
points. In order to secure greater mobility, and not to fall into 
excessive prolixity, I have allowed myself to make tacit use of 
the commutability of antecedents, and of the identification of 
like antecedents, and have not reduced the ways of inference 
and deduction to the smallest number. Those who know my 
small book Begriffsschrift will be able to gather from it how 
the strictest requirements could be forthcoming here as well, 
but also that this would bring with it a notable increase in 
size. 

Frege is correct here in claiming Euclid as his predecessor, insofar 
as Euclid was the first to carry out the idea of an axiomatic system in 
mathematics. But it would have been much better to refer to Aristotle 
(14.02, 14.05), for what Frege offers is an important sharpening of the 
Aristotelian concept of an axiomatic system. His first requirement is 
that all presuppositions should be formulated expressly and without 
gaps. Then he makes an explicit distinction between the laws and the 
methods of inference and deduction, i.e. the rules of inference. This 
is not altogether new (cf. 22.12-22.15, 30.11. § 31, C), but is stated 
with greater clarity than ever before. Finally. Frege can be con- 
trasted with Leibniz, Boole and other earlier writers in his laying 
down of a quite new requirement: 'considerably higher demands 
must be made on the notion of proof than is usual in arithmetic'. 
With that, mathematical logic enters on its second phase. 

38.23 and 38.24, along with the citations from Boole, are texts 
of far-reaching influence on the concept of mathematical logic. In 
this connection two further quotations, dating from 1896, may be 
added. 

38.25 Words such as 'therefore', 'consequently', 'since' 
suggest indeed that inference has been made, but say nothing 
of the principle in accordance with which it has been made, and 
could also be used without misuse of words where there is no 
logically justified inference. In an inquiry which I here have 
in view, the question is not only whether one is convinced of 
the truth of the conclusion, with which one is usually satisfied 

283 



MATHEMATICAL LOGIC 

in mathematics ; but one must also bring to consciousness the 
reason for this conviction and the primitive laws on which it 
rests. Fixed lines on which the deductions must move are 
necessary for this, and such are not provided in ordinary 
language. 

38.26 Inference is conducted in my symbolic system 
(Begriffsschrift) according to a kind of calculation. I do not 
mean this in the narrow sense, as though an algorithm was in 
control, the same as or similar to that of ordinary addition 
and multiplication, but in the sense that the whole is algorith- 
mic, with a complex of rules which so regulate the passage 
from one proposition or from two such to another, that nothing 
takes place but what is in accordance with these rules. My 
aim, therefore, is directed to continuous strictness of proof and 
utmost logical accuracy, along with perspicuity and brevity. 

Frege's program of thorough proof was later carried out in mathe- 
matics by Hilbert with a view to pure formality. The texts can be 
referred to in 0. Becker (38.27). It was Lukasiewicz who applied 
it to logical systems with complete strictness. We shall give an 
example in the chapter on propositional logic (43.45). 



C. METALOGIC 

The idea of a metalogic was an inevitable result of the combination 
of Boole's formalism and Frege's theory of proof. For once formulae 
had been distinguished from rules, and the former treated with strict 
formalism, 'after the fashion of an algorithm' as Frege says, then the 
rules had to be interpreted as meaningful and having content. At 
once the rules are seen as belonging to a different level to the for- 
mulae. The notion of this second level appears first in connection 
with mathematics as that of metamathematics in Hilbert. We cite his 
lecture Die logischen Grundlagen der Malhematik (1923): 

38.28 The basic idea of my theory of proof is this: 
Everything that goes to make up mathematics in the 
accepted sense is strictly formalized, so that mathematics 
proper, or mathematics in the narrower sense, becomes a 
stock of formulae. . . . 

Beyond mathematics proper, formalized in this way, there 
is, so to speak, a new mathematics, a metamathematics, which 
is needed to establish the other securely. In it, by contrast to 
the purely formal ways of inference in mathematics proper, 

284 



PROOF 

inference which has regard to the subject matter is applied, 
though merely to establish the freedom from contradiction of 
the axioms. In this metamathematics we operate with the 
proofs of mathematics proper, these last themselves forming 
the object of the inference that regards the matter. In this 
way the development of the total science of mathematics is 
achieved by a continual exchange which is of two kinds: the 
gaining of new provable formulae from the axioms by means of 
formal inference, and on the other hand the addition of new 
axioms along with the proof of freedom from contradiction by 
means of inference having regard to the matter. 

The axioms and provable propositions, i.e. formulae, which 
arise in this process of exchange, are representations of the 
thoughts which constitute the usual processes of mathematics 
as understood up to now, but they are not themselves truths 
in an absolute sense. It is the insights which are afforded by my 
theory of proof in regard to provability and freedom from 
contradiction which are rather to be viewed as the absolute 
truths. 

This important text goes beyond the bounds of this chapter in 
that it touches not only on proof-theory but also on the concept of 
logic and its relations to mathematics, since Hilbert here limits 
meaningful inference to the proof of freedom from contradiction, 
in accordance with his special philosophy of mathematics. The 
important point for our purpose is chiefly the sharp distinction 
between the formalized, and so in Boole's sense meaningless, 
calculus on the one hand, and the meaningful rules of inference on 
the other. This idea, too, was first expressed by Frege, when he requir- 
ed enumeration of all 'ways of inference and deduction' as distinct 
from axioms (cf. 38.24). But Frege did not think of the axioms and 
theorems as meaningless, however formally he considered them. Here 
on the contrary it is a case of inscriptions considered purely mate- 
rially. 

A new stage in the understanding of formalization has thus been 
reached. The doctrine in Hilbert is, of course, limited to mathematics - 
he speaks of metamathematics. But soon this idea was to be extended 
to logic, and this came about in the Warsaw School. The expression 
'metalogic' first occurs in a paper by Lukasiewicz and Tarski of 
1930 (38.29). 

Parallel to the work of the Warsaw School is that which R. Carnap 
was carrying on in Vienna at the same time. 

We cite now a text of Tarski's, the founder of systematic meta- 
logic. He forms the starting-point for the most recent developments 
which will not be pursued here. 

285 



MATHEMATICAL LOGIC 

Tarski wrote in 1930: 

38.30 Our object in this communication is to define the 
meaning, and to establish the elementary properties, of some 
important concepts belonging to the methodology of the deductive 
sciences, which, following Hilbert, it is customary to call 
metamathematics . 

Formalized deductive disciplines form the field of research of 
metamathematics roughly in the same sense in which spatial 
entities form the field of research in geometry. These discip- 
lines are regarded, from the standpoint of metamathematics, 
as sets of sentences. Those sentences which (following a sug- 
gestion of S. Lesniewski) are also called meaningful sentences, 
are themselves regarded as certain inscriptions of a well- 
defined form. The set of all sentences is here denoted by the 
symbol i S i . From the sentences of any set X certain other 
sentences can be obtained by means of certain operations 
called rules of inference. These sentences are called the conse- 
quences of the set X. The set of all consequences is denoted by 
the symbol l Cn {X)\ 

An exact definition of the two concepts, of sentence and of 
consequence, can be given only in those branches of meta- 
mathematics in which the field of investigation is a concrete 
formalized discipline. On account of the generality of the 
present considerations, however, these concepts will here be 
regarded as primitive and will be characterized by means of 
a series of axioms. 



§39. THE CONCEPT OF LOGIC 

As has been seen above (§ 38) Boole (38.17), Peirce (38.19) and 
with them the other mathematical logicians of the 19th century 
considered logic to be a branch of mathematics, this last being 
described not with reference to its object but its method, the 
application of a calculus. However, at the end of the 19th century 
there arose considerable disagreement about the relationship of 
logic to mathematics, a disagreement which at the same time 
concerned the answer to the question whether logic can be deve- 
loped purely formally as a system of symbols, or whether it necessa- 
rily involves an interpretation of the symbols. So there were two 
problems, but both concerned with the concept of logic. Three main 
positions took shape: the logistic, the formalistic (not in the sense 
in which 'formalism' is used in the last and in subsequent sections) 

286 



CONCEPT OF LOGIC 

and the intuitionistic. We shall illustrate their main features with 
some texts. 

A. THE LOGISTIC POSITION 

On the logistic position there is no essential distinction between 
logic and mathematics, inasmuch as mathematics can be developed 
out of logic; more exactly, inasmuch as all mathematical terms 
can be defined by logical ones, and all mathematical theorems 
can be deduced from true logical axioms. Frege is the originator of 
this line of thought, which attained its fullest development in the 
Principia Malhematica of Whitehead and Russell, written precisely 
to provide a thorough proof of the logistic thesis. 

1. Frege: semantics 

Frege's theory of logic is closely connected with his semantics 
(a word which we always use here in the sense of Morris (5.01), not 
in Tarski's technical sense). On this point we shall here recall 
briefly only that logic for Frege was not a game with symbols but a 
science of objective thoughts (Gedanken), i.e. of ideal propositions 
(and so of lecta in the sense of 19.04ff.). The premisses must be true, 
formalism is only a means. To begin with, we give a text about the 
first point: 

39.01 By the word 'sentence' (Satz) I mean a sign which is 
normally composite, regardless of whether the parts are 
spoken words or written signs. This sign must naturally have 
a sense (Sinn). I shall here only consider sentences in which we 
assert or state something. We can translate a sentence into 
another language. In the other language the sentence is 
different from the original one, since it consists of different 
components (words) differently compounded; but if the 
translation is correct, it expresses the same sense. And the 
sense is properly just that which matters to us. The sentence 
has a value for us through the sense which we apprehend in 
it, and which we recognize as the same in the translation too. 
This sense I call 'thought' (Gedanke). What w r e prove is not 
the sentence but the thought. And it makes no difference what 
language we use for that purpose. In mathematics people 
speak indeed of a proof of a Lehrsaiz when they understand 
by the word Satz what I call 'thought' - or perhaps they do not 
sufficiently distinguish between the verbal or symbolic 
expression and the thought expressed. But for clarity it is 
better to make this distinction. The thought is not perceptible 
to the senses, but we give it an audible or visible represen- 

287 



MATHEMATICAL LOGIC 

tative in the sentence. Hence I say 'theorem' rather than 
'sentence', 'axiom' rather than 'primitive sentence', and by 
theorems and axioms I understand true thoughts. This further 
implies that thoughts are not something subjective, the 
product of our mental activity; for the thought, such as we 
have in the theorem of Pythagoras, is the same for everyone, 
and its truth is quite independent of whether it is or is not 
thought by this or that man. Thinking is to be viewed not as 
the production of thought but as its understanding. 

Here, in another terminology, we have exactly the Stoic doctrine 
that logic deals with lecta, and the third scholastic view (28.17) 
according to which propositions stand for ideal structures. 

On the question of the truth of premisses, Frege says : 

39.02 Nothing at all can be deduced from false premisses. 
A mere thought which is not accepted as true, cannot be a 
premiss. Only when I have accepted a thought as true can it 
be a premiss for me; mere hypotheses cannot be used as 
premisses. Of course, I can ask what consequences follow from 
the supposition that A is true without having accepted the 
truth of A ; but the result then involves the condition : if 
A is true. But that is only to say that A is not a premiss, since 
a true premiss does not occur in the judgment inferred. 

Frege thus holds a kind of absolutest doctrine closely approximat- 
ing to the Aristotelian theory of a7c68ei£i<; (14.02) but apparently 
still more radical. 

We append now a characteristic text about the use of quotation- 
marks, in which Frege's high degree of exactness finds expression - a 
degree that has been too seldom attained since. 

39.03 People may perhaps wonder about the frequent use 
of quotation-marks ; I use them to distinguish the cases where 
I am speaking of the symbol itself, from those where I am 
speaking of what it stands for. This may seem very pedantic, 
but I consider it necessary. It is extraordinary how an inexact 
manner of speaking and writing, which was originally perhaps 
used only for convenience, can in the end lead thought 
astray after one has ceased to notice it. Thus it has come about 
that numerals are taken for numbers, names for what they 
name, what is merely auxiliary for the proper object of 
arithmetic. Such experiences teach us how necessary it is to 
demand exactness in ways of talking and writing. 

288 



CONCEPT OF LOGIC 
2. Frege: Logic and Mathematics 

39.04 Under the name 'formal theory' I shall here consider 
two modes of conception, of which I subscribe to the first and 
endeavour to refute the second. The first says that all arith- 
metical propositions can, and hence should, be deduced from 
definitions alone by purely logical means. . . . Out of all the 
reasons which support this view I shall here adduce only one, 
which is based on the comprehensive applicability of arith- 
metical doctrines. One can in fact number pretty well every- 
thing that can be an object of thought: the ideal as well as the 
real, concepts and things, the temporal and the spatial, events 
and bodies, methods as well as propositions; numbers them- 
selves can be in turn numbered. Nothing is actually required 
beyond a certain definiteness of delimitation, a certain logical 
completeness. From this there may be gathered no less than 
that the primitive propositions on which arithmetic is based, 
are not to be drawn from a narrow domain to the special 
character of which they give expression, as the axioms of 
geometry express the special character of the spatial domain; 
rather must those primitive propositions extend to everything 
thinkable, and a proposition of this most universal kind is 
rightly to be ascribed to logic. 

From this logical or formal character of arithmetic I draw 
some conclusions. 

First: no sharp boundary between logic and arithmetic 
is to be drawn; considered from a scientific point of view 
both constitute a single science. If the most universal primi- 
tive propositions and perhaps their immediate consequences 
are attributed to logic, and the further development to 
arithmetic, it is like wanting to detach a special science of 
axioms from geometry. Yet the partitioning of the whole 
domain of knowledge among the sciences is determined not 
only by theoretical but also by practical considerations, so 
that I do not wish to say anything against a certain practical 
separation. But it must not become a breach as is now the 
case to the detriment of both. If this formal theory is correct, 
logic cannot be so fruitless as it may appear to a superficial 
consideration - of which logicians are not guiltless. And there 
is no need for that attitude of reserve on the part of many 
mathematicians towards any philosophic justification of 
whatever is real, at least insofar as it extends to logic. This 

289 



MATHEMATICAL LOGIC 

science is capable of no less exactness than mathematics 
itself. On the other hand logicians may be reminded that they 
cannot learn to know their own science thoroughly if they 
do not trouble themselves about arithmetic. 

39.05 My second conclusion is that there is no special 
arithmetical kind of inference such that it cannot be reduced 
to the common inference of logic. 

39.06 My third conclusion concerns definitions, as my 
second concerned kinds of inference. In every definition 
something has to presupposed as known, by means of which 
one explains what is to be understood by a name or symbol. 
An angle cannot be well defined without presupposing know- 
ledge of a straight line. Now that on which a definition is 
based may itself be defined; but in the last resort one must 
always come to something indefinable, which has to be 
recognized as simple and incapable of further resolution. And 
the properties which belong to these foundation stones of 
science, contain its whole content in embryo. In geometry 
these properties are expressed in the axioms, to the extent 
that these are independent of one another. Now it is clear 
that the boundaries of a science are determined by the 
nature of its foundation stones. If, as in geometry, we are 
originally concerned with spatial structures, the science, too, 
will be limited to what is spatial. Since then arithmetic is to 
be independent of all particular properties of things, that 
must hold for its foundations : they must be of a purely logical 
kind. The conclusion follows that everything arithmetical 
is to be reduced by definitions to what is logical. 

3. Russell 

Frege's postulates were first taken up by Giuseppe Peano - though 
without direct dependence on Frege - then by Bertrand Russell. 
The latter extended the logistic thesis to geometry and mathematical 
disciplines in general. 

39.07 The general doctrine that all mathematics is deduc- 
tion by logical principles from logical principles was strongly 
advocated by Leibniz. . . . But owing partly to a faulty logic, 
partly to belief in the logical necessity of Euclidean Geometry, 
he was led into hopeless errors. . . . The actual propositions 
of Euclid, for example, do not follow from the principles of 
logic alone; .... But since the growth of non-Euclidean 
Geometry, it has appeared that pure mathematics has no 

290 



CONCEPT OF LOGIC 

concern with the question whether the axioms and propositions 
of Euclid hold of actual space or not: this is a question for 

applied mathematics, to be decided, so far as any decision is 
possible, by experiment and observation. What pure mathe- 
matics asserts is merely that the Euclidean propositions follow 
from the Euclidean axioms - i.e. it asserts an implication: 
any space which has such and such properties has also such 
and such other properties. Thus, as dealt with in pure mathe- 
matics, the Euclidean and non-Euclidean Geometries are 
equally true: in each nothing is affirmed except implica- 
tions. . . . 

39.08 Thus pure mathematics must contain no indefinables 
except logical constants, and consequently no premisses, or 
indemonstrable propositions, but such as are concerned 
exclusively with logical constants and with variables. It is 
precisely this that distinguishes pure from applied mathe- 
matics. 

How and to what extent this program was carried out, cannot 
here be pursued. Reference may be made to Becker (39.09). In 
conclusiou we should like only to illustrate Frege's definition of 
number by means of purely logical concepts, especially with a view 
to comparing it with a similar discovery in the Indian logic of the 
17th century (54.17). 

4. Frege: number 

39.10 To illuminate matters it will be good to consider 
number in connection with a judgment where its primitive 
manner of application occurs. If when I see the same outward 
appearances I can say with the same truth: 'this is a group of 
trees' and 'these are five trees' or 'here are four companies' 
and 'here are 500 men', no difference is made to the individual 
or to the whole, the aggregate, but to my naming. But this 
is only the sign of the substitution of one concept by another. 
This suggests an answer to the first question of the previous 
paragraph, that number involves a statement about a concept. 
This is perhaps most evident for the number 0. When I say: 
'Venus has moons', there is no moon or aggregate of moons 
there about which anything can be said; but to the concept 
'moon of Venus' there is attributed a property, viz. that of 
comprising nothing under it. When I say: 'the emperor's 
carriage is drawn by four horses', I apply the number four to 
the concept 'horse which draws the emperor's carriage'. . . . 

291 



MATHEMATICAL LOGIC 

39.11 Among the properties which are predicated of a 
concept I do not, of course, understand the notes which make 
up the concept. These are properties of the things which fall 
under the concept, not of the concept. Thus 'right-angled' 
is not a property of the concept 'right-angled triangle'; but 
the proposition that there is no right-angled, rectilineal, 
equilateral triangle, states a property of the concept 'right- 
angled, rectilineal, equilateral triangle', attributing to it the 
number 0. 

39.12 In this respect existence is like number. The affir- 
mation of existence is nothing else than the denial of the 
number 0. Since existence is a property of the concept, the 
ontological proof of the existence of God fails of its purpose 

It would also be false to deny that existence and unicity 
can ever be notes of concepts. They are only not notes of that 
concept to which the manner of speech might lead one to 
ascribe them. E.g. when all concepts belonging only to one 
object are collected under one concept, uniqueness is a note of 
this concept. The concept 'moon of the earth', for instance, 
would fall under it, but not the so-called heavenly body. 
Thus a concept can be allowed to fall under a higher one, 
under a concept, so to speak, of second order. But this rela- 
tionship is not to be confused with that of subordination. 

Frege's definition of number was later interpreted by Russell 
extensionally, when he took numbers as classes of classes (39.13). 

B. FORMALISM 

The formalists, too, see no essential difference between logical 
and mathematical formulae, but they understand both forma- 
listically and think of the single system composed of them as a 
system of symbols. Evidence and truth of the axioms have no part 
to play: but freedom from contradiction is everything. The founder 
of formalism is David Hilbert, the essentials of whose thought on 
logic is contained in the text given earlier (38.28). Here we add only 
a brief passage from a letter to Frege in 1899 or 1900: 

39.14 You write: 'From the truth of the axioms it follows 
that they do not contradict one another'. I was very interested 
to read this particular sentence of yours, because for my part, 
ever since I have been thinking, writing and lecturing about 
such matters, I have been accustomed to say just the reverse: 
if the arbitrarily posited axioms are not in mutual (sic) con- 

292 



CONCEPT OF LOGIC 

tradiction with the totality of their consequences, then they 
are true - the things defined by the axioms exist. That for 
me is the criterion of truth and existence. 

For the rest, it is not easy to find texts to illustrate Hilbert's 
thought before 1930; for that and the later development Becker 
may again be consulted (39.15). 

It should be noted that formalism has been very important for the 
concept of logic, quite apart from its value as a theory. Logic having 
been previously viewed as a calculus, it is henceforth ever more and 
more transposed onto the level of metalogic. After Hilbert, it is not 
the formulae themselves but the rules of operation by which they 
are formed and derived that are more and more made the object of 
logical investigation. 

C. INTUITIONISM 

By contrast to the logisticians and formalists, the intuition ists 
make a sharp distinction between logic and mathematics. Mathe- 
matics is not, for them, a set of formulae, but primarily a mental 
activity the results of which are subsequently communicable by 
means of language. In language, as used by mathematicians, 
certain regularities are observed, and this leads to the development 
of a logic. Thus logic is not presupposed by, but abstracted from 
mathematics. Once that has been done, it can then be formalized, 
but this is a matter of secondary importance. * 

Intuitionism has a fairly long history in mathematics: L. Kron- 
ecker and H. Poincare are precursors; H. Weyl is reckoned a 
'semi-intuitionist'. But L. E. J. Brouwer ranks as the founder of 
the school, and intuitionistic logic was first properly formulated 
(and formalized) by A. Heyting in 1930. 

From the standpoint of formal logic it is to be noted that the 
intuitionists, as they themselves say, admit the principle of tertium 
exclusion only under certain limitations. In this respect their 
doctrine belongs to those 'heterodox' logics of which we shall speak 
in § 49. 

We give one text from Heyting and one from Brouwer : 

39.16 Intuitionistic mathematics is an activity of thought. 
and every language - even the formalistic - is for it only a 
means of communication. It is impossible in principle to 
establish a system of formulae that would have the same 
value as intuitionistic mathematics, since it is impossible to 

* Special thanks are due to Prof. E. W. Beth for much information in this 
connection, as generally for his help with the composition of this fifth part. 

293 



MATHEMATICAL LOGIC 

reduce the possibilities of thought to a finite number of 
rules that thought can previously lay down. The endeavour 
to reproduce the most important parts of mathematics in a 
language of formulae is justified exclusively by the great 
conciseness and defmiteness of this last as compared with 
customary languages, properties which fit it to facilitate 
penetration of the intuitionistic concepts and their applica- 
tion in research. 

For constructing mathematics the statement of universally 
valid logical laws is not necessary. These laws are found as it 
were anew in every individual case to be valid for the mathe- 
matical system under consideration. But linguistic com- 
munication moulded according to the needs of everyday life 
proceeds according to the form of logical laws which it 
presupposes as given. A language which imitated the process 
of intuitionistic mathematics step by step would so diverge in 
all its parts from the usual pattern that it would have to 
surrender again all the useful properties mentioned above. 
These considerations have led me to begin the formalization 
of intuitionistic mathematics once again with a propositional 
calculus. 

The formulae of the formalistic systems come into being by 
the application of a finite number of rules of operation to a 
finite number of axioms. Besides 'constant' symbols they also 
contain variables. The relationship between this system and 
mathematics is this, that on a determinate interpretation 
of the constants and under certain restrictions on substitu- 
tion for variables every formula expresses a correct mathe- 
matical proposition. (E.g. in the propositional calculus the 
variables must be replaced only by senseful mathematical 
sentences.) If the system is so constructed as to fulfil the 
last-mentioned requirement, its freedom from contradiction 
is thereby guaranteed, in the sense that it cannot contain 
any formula which would express a contradictory proposition 
on that interpretation. 

The formalistic system can also be considered mathemati- 
cally for its own sake, without reference to any interpretation. 
Freedom from contradiction then takes on a new meaning 
inasmuch as contradiction is defined as a definite formula; 
for us this method of treatment is less to the fore than the 
other. But here questions come in about the independence and 
completeness of the axiom-system. 

294 



CONCEPT OF LOGIC 

39.17 The differences about the Tightness of the new 
formalistic foundations and the new intuitionistic construc- 
tion of mathematics will be removed, and the choice between 
the two methods of operation reduced to a matter of taste, 
as soon as the following intuitions (Einsichten) have been 
generally grasped. They primarily concern formalism, but 
were first formulated in intuitionist literature. This grasp 
is only a matter of time, since they are results purely of 
reflection, containing nothing disputable, and necessarily 
acknowledged by everyone who has once understood them. 
Of the four intuitions this understanding and acknowledge- 
ment has so far been attained for two in the formalistic 
literature. Once the same state of affairs has been reached for 
the other two, an end will have been put to disputes about 
foundations in mathematics. 

FIRST INTUITION. The distinction between the formalistic 
endeavours to construct the 'mathematical slock of formulae' 
(formalistic idea of mathematics) and an intuitive (meaningful) 
theory of the laws of this construction, as also the understanding 
that for the last theory the intuitionistic mathematics of the set 
of natural numbers is indispensable. 

SECOND INTUITION. The rejection of the thoughtless 
application of the logical theorem of tertium exclusum, as also 
the awareness first, that investigation of the credentials and 
domain of validity of the said theorem constitutes an essential 
object of mathematical foundational research; second, that this 
domain of validity in intuitive (meaningful) mathematics com- 
prises only finite systems. 

THIRD INTUITION. The identification of the theorem of 
tertium exclusum with the principle of the solubility of every 
mathematical problem. 

FOURTH INTUITION. The awareness that the (meaningful) 
justification of formalistic mathematics through proof of its 
freedom from contradiction involves a vicious circle, since this 
justification depends on the (meaningful) correctness of the 
proposition that the correctness of a proposition follows from 
the freedom from contradiction of this proposition, i.e. from the 
(meaningful) correctness of the theorem of tertium exclusum. 



295 



II. THE FIRST PERIOD 



§40. THE BOOLEAN CALCULUS 

The system of mathematical logic inaugurated by Boole in 1847 
holds a special place in history in that it admits of two interpreta- / 
tions, in class-logic and propositional logic. In this section we shall ^ 
consider the abstract calculus itself and its classical interpretation, 
reserving the propositional interpretation to the following section. 

The growth of Boole's calculus can be summarized as follows: 
De Morgan is its precursor (though his chief work was published 
contemporaneously with Boole's in 1847); Boole set out the main 
lines of the system in that year; but his exposition lacks the concept 
of the logical sum which first appears in Peirce (1867), Schroder 
(1877), and Jevons (1890), as also the concept of inclusion, originally 
introduced by Gergonne (1816) and clearly formulated by Peirce in 
1870. Schroder's system (1890) ranks as the completion of this 
growth, though perhaps Peano's (1899) may here be counted as the 
real close. 

A. DE MORGAN 

Boole's calculus emerged in a way from the 'classical' endeavours 
to broaden the Aristotelian syllogistic (36.15 f.). This can be most 
clearly seen from the syllogistic of Augustus de Morgan. 

40.01 I shall now proceed to an enlarged view of the 
proposition, and to the structure of a notation proper to repre- 
sent its different cases. 

As usual, let the universal affirmative be denoted by A, 
the particular affirmative by /, the universal negative by E, 
and the particular negative by O. This is the extent of the 
common symbolic expression of propositions: I propose to 
make the following additions for this work. Let one particular 
choice of order, as to subject and predicate, be supposed 
established as a standard of reference. As to the letters X, F,. 
Z, let the order always be that of the alphabet, IF, YZ, XZ 
Let x, y, z, be the contrary names of X, F, Z; and let the 
same order be adopted in the standard of reference. Let the 
four forms when choice is made of an X, F, Z, be denoted by 
A t , E , /,, O,; but when the choice is made from the contraries, 
let them be denoted by A', E\ /', O' . Thus with reference to 
Y and Z, "Every Y is Z" is the A t of that pair and order: 
while "Every y is z" is the A'. I should recommend ^and^' 

296 



BOOLEAN CALCULUS 

to be called the sub-A and the super- A of the pair and order 
in question: the helps which this will give the memory will 
presently be very apparent. And the same of I t and /', etc. 
Let the following abbreviations be employed; - X) Y 
means "Every X is Y". X. Y means "No X is Y". X:Y means 
"Some Xs are not Ys". XY means "Some Xs are Fs". 

Later, De Morgan developed a different symbolism. We give it- 
description and a comparative table, from a paper of 18. r jf): 

40.02 Let the subject and predicate, when specified, be 
written before and after the symbols of quantity. Let the 
enclosing parenthesis, as in X) or (X, denote that the name- 
symbol X, which would be enclosed if the oval were completed, 
enters universally. Let an excluding parenthesis, as in )X 
or X(, signify that the name-symbol enters particularly. Let 
an even number of dots, or none at all, inserted between 
the parentheses, denote affirmation or agreement; let an odd 
number, usually one, denote negation or non-agreement. 



40.03 


Universals 




Former 


Notation of Both. 


Notation 


Proposition 


memoir. 


my Work on 


now 


expressed in 




Logic. 


proposed. 


common 
language. 


A 


A 1 X)Y 


X))Y 


Every X is F 


a 


A 1 x ) y or 


x )) y or 


Every F is X 




Y)X 


X(( Y 




E 


E 1 X)yov 


X))yov 


No X is F 




X . Y 


X).( Y 




e 


E 1 x ) For 


x))Yor 


Everything is 




x.y 


X(.) Y 


X or F o rboth 




Particulars 




I 


I t XY 


X()Y 


Some Xs are Fs 


i 


I 1 xy 


x()yor 


Some things 






X)(Y 


are neither .Ys 
nor Fs 





O x Xy or 


XQyor 


Some Xs are 




X : F 


X (.( Y 


not Fs 





O 1 xY or 


x () F or 


Some Fs are 




F : X 


X).)Y 


not Xs 



•297 



MATHEMATICAL LOGIC 
B. BOOLE 

Boole, who was the first to outline clearly the program of mathe- 
matical logic, was also the first to achieve a partial execution. In 
this respect there is a great likeness between his relationship to 
Leibniz and that of Aristotle to Plato. For with Boole as with 
Aristotle we find not only ideas but a system. 

This system of Boole's can be described thus: it is in the first 
r^ace closely allied to arithmetic, in that it uses only arithmetical 
symbols and has only one law that diverges from those of arith- 
metic, viz. x n = x. All its procedures are taken over from simple 
algebra; Boole has no conscious awareness of purely logical methods 
(even of those which are intuitively used in algebra), e.g. of the rules 
of detachment and substitution. As a matter of fact, even the basic 
law mentioned makes very little difference to the algebraic character 
of his system - which is algebra limited to the numbers and 1. 

Boole's mathematicism goes so far - and this is the second main 
characteristic of his doctrine - that he introduces symbols and 
procedures which admit of no logical interpretation, or only a com- 
plicated and scarcely interesting one. Thus we meet with subtrac- 
tion and division and numbers greater than 1. 

From the logical point of view it is to be noted that disjunction 
(symbolized by l x + y') is taken as exclusive, and that inclusion is 
expressed by means of equality. Both lead to difficulties and 
unnecessary complications; both are the result of the tendency 
to mathematicize. 

A third and special characteristic is that the system possesses 
two interpretations, in classical and propositional logic. 

Altogether, in spite of its defects, Boolean algebra is a very 
successful piece of logic. Boole resembles Aristotle both in point of 
originality and fruitfulness ; indeed it is hard to name another 
logician, besides Frege, who has possessed these qualities to the 
same degree, after the founder. 

1. Symbolism and basic concepts 

40.04 Proposition I. All the operations of the Language, 
as an instrument of reasoning, may be conducted by a system 
of signs composed of the following elements, viz. : 

1st. Literal symbols as x, y, etc., representing things as 
subjects of our conceptions. 

2nd. Signs of operation, as +, -, x, standing for those 
operations of the mind by which the conceptions of things 
are combined or resolved so as to form new conceptions 
involving the same elements. 

3rd. The sign of identity, =. 

298 



BOOLEAN CALCULUS 

And these symbols of Logic are in their use subject to 
definite laws, partly agreeing with and partly differing from 
the laws of the corresponding symbols in the science of 
Algebra. 

40.05 Let us employ the symbol 1 or unity, to represent 
the Universe, and let us understand it as comprehending 
every conceivable class of objects whether actually existing 
or not, it being premised that the same individual may La 
found in more than one class, inasmuch as it may possess 
more than one quality in common with other individuals. 
Let us employ the letters X, Y, Z, to represent the individual 
members of classes, X applying to every member of one 
class, as members of that particular class, and Y to every 
member of another class as members of such class, and so on, 
according to the received language of treatises on Logic. 

Further let us conceive a class of symbols x, y, z, possessed 
of the following character. 

The symbol x operating upon any subject comprehending 
individuals or classes, shall be supposed to select from that 
subject all the Xs which it contains. In like manner the 
symbol y, operating upon any subject, shall be supposed to 
select from it all individuals of the class Y which are com- 
prised in it, and so on. 

When no subject is expressed, we shall suppose 1 (the 
Universe) to be the subject understood, so that we shall 
have 

40.051 x = x (1), 

the meaning of either term being the selection from the 
Universe of all the Xs which it contains, and the result of 
the operation being in common language, the class X, i.e. the 
class of which each member is an X . 

From these premises it will follow, that the product xy will 
represent, in succession, the selection of the class Y, and the 
selection from the class Y of such individuals of the class X 
as are contained in it, the result being the class whose members 
are both Xs and Fs. . . . 

From the nature of the operation which the symbols x, 
y, z, are conceived to represent, we shall designate them as 
elective symbols. An expression in which they are involved 
will be called an elective function, and an equation of which 
the members are elective functions, will be termed an elective 
equation. . . . 

299 



MATHEMATICAL LOGIC 

1st. The result of an act of election is independent of the 
grouping or classification of the subject. 

Thus it is indifferent whether from a group of objects 
considered as a whole, we select the class X, or whether we 
divide the group into two parts, select the Xs from them 
separately, and then connect the results in one aggregate 
conception. 

We may express this law mathematically by the equation 

(40.052) x (u + v) = xu + xv, 

u + v representing the undivided subject, and u and v the 
component parts of it. 

2nd. It is indifferent in what order two successive acts of 
election are performed. 

Whether from the class of animals we select sheep, and 
from the sheep those which are horned, or whether from the 
class of animals we select the horned, and from these such as 
are sheep, the result is unaffected. In either case we arrive 
at the class of horned sheep. 

The symbolical expression of this law is 

(40.053) xy = yx. 

3rd. The result of a given act of election performed twice, 
or any number of times in succession, is the result of the 
same act performed once. . . . Thus we have 

(40.054) xx = x, 
or x 2 = x: 

and supposing the same operation to be n times performed, 
we have 

(40.055) x n =x, 

which is the mathematical expression of the law above 
stated. 

The laws we have established under . . . symbolical forms . . . 
are sufficient for the base of a Calculus. From the first of 
these it appears that elective symbols are distributive, from 
the second that they are commutative; properties which they 
possess in common with symbols of quantity, and in virtue 
of which, all the processes of common algebra are applicable 
to the present system. The one and sufficient axiom involved 
in this application is that equivalent operations performed 
upon equivalent subjects produce equivalent results. 

The third law ... we shall denominate the index law. It 
is peculiar to elective symbols. 

300 



BOOLEAN CALCULUS 
2. Applications 

We now give two examples of the application of these principles 
in Boole's work. The first concerns the law of contradiction. 

40.06 That axiom of metaphysicians which is termed the 
principle of contradiction, and which affirms that it is impos- 
sible for any being to possess a quality, and at the same time 
not to possess it, is a consequence of the fundamental law 
of thought, whose expression is x 2 = x. 

Let us write this equation in the form 

(40.061) x-x 2 = 
whence we have 

(40.062) x(l -x) = 0; (1) 

both these transformations being justified by the axiomatic 
laws of contradiction and transposition. . . . Let us for simpli- 
city of conception, give to the symbol x the particular inter- 
pretation of men, then 1 - x will represent the class of 'not- 
men'. . . . Now the formal product of the expressions of two 
classes represents that class of individuals which is common 
to them both. . . . Hence x (1 - x) will represent the class 
whose members are at once 'men', and 'not-men', and the 
equation (1) thus expresses the principle, that a class whose 
members are at the same time men and not men does not exist. 
In other words, that it is impossible for the same individual to 
be at the same time a man and not a man. Now let the meaning 
of the symbol x be extended from the representing of 'men', 
to that of any class of beings characterized by the possession of 
any quality whatever; and the equation (1) will then express 
that it is impossible for a being to possess a quality and not 
to possess that quality at the same time. But this is identically 
that 'principle of contradiction' which Aristotle has de- 
scribed as the fundamental axiom of all philosophy. . . . 

The above interpretation has been introduced not on 
account of its immediate value in the present system, but as 
an illustration of a significant fact in the philosophy of the 
intellectual powers, viz., that what has been commonly re- 
garded as the fundamental axiom of metaphysics is but the 
consequence of a law of thought, mathematical in its form. 

The second example is taken from the application of Boole's 
methods in the domain of syllogistic. 

40.07 The equation by which we express any Proposition 
concerning the classes X and Y, is an equation between the 

301 



MATHEMATICAL LOGIC 

symbols x and y, and the equation by which we express any 
proposition concerning the classes Y and Z, is an equation 
between the symbols y and z. If from two such equations we 
eliminate y, the result, if it do not vanish, will be an equation 
between x and z, and will be interpretable into a Proposition 
concerning the classes X and Z. And it will then constitute 
the third member, or Conclusion, of a Syllogism, of which 
the two given Propositions are the premises. 

The result of the elimination of y from the equations 

ay + b = 0, 

(14) 

a' y - b' = 0, 
is the equation ab' - a' b = (15). 

40.08 Ex(ample). AA, Fig. 1, and by mutation of premises 
(change of order), AA, Fig. 4. 

All Ys are Xs, y (1 - x) = 0, or (1 - x) y = 0, 
All Zs are Ys, z (1 - y) = 0, or zy - z = 0. 
Eliminating y by (15) we have 
z (1 - x) = 0, 
All Zs are Xs. 

In both these texts Boolean methods are being applied to tradi- 
tional problems, involving logical relationships between two objects 
(classes, propositions). But the interesting thing about this calculus 
for our history is that it is applicable to more than two objects, 
so that it oversteps the limits of the 'classical' logic. An instance 
is given later (41.03). 

C. THE LOGICAL SUM 

The original Boolean calculus had two main defects from the 
logical point of view, both occasioned by its extreme mathemati- 
cism; disjunction was treated as exclusive, and there was no symbol 
to hand for inclusion, though that is fundamental in logic. The first 
defect was remedied by Jevons, who was strongly opposed to this 
mathematicism and introduced non-exclusive disjunction. 

49.09 There are no such operations as addition and sub- 
traction in pure logic. . . 

40.10 Now addition, subtraction, multiplication, and divi- 
sion, are alike true as modes of reasoning in numbers, where 
we have the logical condition of a unit as a constant restriction. 
But addition and subtraction do not exist, and do not give 
true results in pure logic, free from the conditions of number. 

302 



BOOLEAN CALCULUS 

For instance take the logical proposition - 
A+B+C=A+D+E 
meaning what is either A or B or C is either A or D or E, and 
vice versa. There being no exterior restriction of meaning 
whatever, except that some terms must always have the same 
meaning, we do not know which of A, D, E, is B, nor which 
is C;. . . . The proposition alone gives us no such information. 

Much clearer is Charles S. Peirce, also an opponent of Boole's 
mathematicism (1867). He uses an appropriate though still primitive 
symbolism. 

40.11 Let the sign of equality with a comma beneath it 
express numerical identity. . . . Let a -t b denote all the 
individuals contained under a and b together. The operation 
here performed will differ from arithmetical addition in two 
respects : first, that it has reference to identity, not to equality, 
and second, that what is common to a and b is not taken into 
account twice over, as it would be in arithmetic. The first of 
these differences, however, amounts to nothing, inasmuch as 
the sign of identity would indicate the distinction in which it 
is founded; and therefore we may say that 

(1) If No a is b a -t 6 ^ a + b. 
It is plain that 

(2) a -t a f fl 

and also, that the process denoted by -t, and which I shall call 
the process of logical addition, is both commutative and 
associative. That is to say 

(3) a -b b =? b -t a 
and 

(4) (a h? b) -t c 7= a -t (b -t c). 

This is the third time that non-exclusive disjunction is discovered, 
cf. Galen (20.18) and Burleigh (30.20). 

A symbolism quite different from that of mathematics is first met 
with in Peano (41.20). 

D. INCLUSION 

The introduction of the concept of inclusion and a symbol for it 
has a fairly long history. The modern symbol appears thirty years 
before Boole's Analysis and quite independently of his calculus 
in J. D. Gergonne's Essai de dialedique ralionelle, 1816/17. (The 
parentheses enclosing the italic capitals in this text are Gergonne's.) 

303 



MATHEMATICAL LOGIC 

40.12 We have chosen the signs to characterize these rela- 
tions in the way which seems best for linking the sign to the 
thing signified, and this is an endeavour which we think of 
some importance, however puerile it may appear at first. The 
letter (H), initial letter of the word Hors (outside) designates 
the system of two ideas completely outside one another, as are 
the two vertical strokes of this letter. These two strokes can 
next be considered as crossed to form the letter (X) intended 
to recall the system of two ideas which, as it were, somehow 
intersect. Finally the two strokes can be identified so as to 
form the letter (/) which we use to represent the system of 
two ideas which exactly coincide with one another; this letter 
is, moreover, the initial letter of the word Identity, the denomi- 
nation suitable to the kind of relation in question. It may also 
be noted that the three letters (H, X, I) are symmetrical, 
like the relations they are intended to recall, so that they 
are not liable to change their appearance by being reversed. 
But this is not the case with the letter (67) which on being 
reversed changes into (j); hence we have reserved this letter 
to recall a relation in which the two ideas play different parts, 
a relation which is not at all reciprocal. This letter is, moreover, 
the initial letter common to both of the words Containing 
and Contained, which well express the relative situation of the 
two ideas. 

But it was Charles S. Peirce who in 1870 systematically elaborated 
the concept of inclusion. 

40.13 Inclusion in or being as small as is a transitive relation. 
The consequence holds that 

If x-<y, 

and y — < z, 

then x — < z. 
(Footnote) I use the sign — < in place of ^. My reasons for 
not liking the latter sign are that it cannot be written rapidly 
enough, and that it seems to represent the relation it expresses 
as being compounded of two others which in reality are 
complications of this. It is universally admitted that a higher 
conception is logically more simple than a lower one under it. 
Whence it follows from the relations of extension and com- 
prehension, that in any state of information a broader concept 
is more simple than a narrower one included under it. Now all 
equality is inclusion in, but the converse is not true; hence 

304 



BOOLEAN CALCULUS 

inclusion in is a wider concept than equality, and therefore 
logically a simpler one. On the same principle, inclusion is also 
simpler than being less than. The sign ^ seems to involve a 
definition by enumeration; and such a definition offends 
against the laws of definition. 

Schroder introduces and explains the symbol of inclusion from 
the start: 

40.14 Examples of categorical judgements of the simplest 
kind are propositions accepted as true in chemistry: 

'Gold is metal' - 'Common salt is sodium chloride'. - 

Even to these we can very easily link the basic contrasts 
needed in our science. 

Both statements have the same copula. . . . Yet the factual 
relation between the subject and predicate of the statement is 
essentially different in the first and in the second case, insofar 
as conversely metal is not always gold, while, sodium chloride is 
also common salt. This difference is not expressed in a way 
apparent to the eye in the original statements. 

If it is now desired to exhibit the factual relation between 

subject and predicate by a relative symbol more exactly than 

those statements do, a symbol must be chosen for the first 

example different from that for the second. One might write : 

gold (^ metal common salt = sodium chloride 

40.15 The other symbol Q can be read . . . 'subordinated'. It 
is called the symbol of subordination and a statement such as 

aQb, 
a 'subordination'. The symbol is shaped similarly to, and to 
some extent in imitation of, the 'inequality symbol' of 
arithmetic, viz. the symbol < for 'less [than]'. As is well 
known, this can be read backwards as 'greater', >, and it is 
easily impressed on the memory together with its meaning if 
one bears in mind that the symbol broadens from the smaller 
to the larger value, or points from the larger value towards 
the smaller. Analogously, our symbol of subordination, when 
read backwards in the reversed position, ^), i.e. reading again 
from left to right, will mean ' super ordinate d\ The original 
subordination may also be written backwards as a superordi- 
nation' : 

and this expression means just the same as the original one. 

305 



MATHEMATICAL VARIETY OF LOGIC 

40.16 The copula 'is' is sometimes used to express one, 
sometimes the other of the relations which we have shown 
by means of the symbols (2 and =. For its exhibition a symbol 
composed of both the two last, =£, is chiefly recommended, as 
being immediately, and so to say of itself, intelligible, and 
readily memorizable. In fullest detail, this symbol is to be 
read as 'subordinated or equaV . . . . 

A statement of the form 

a=C° 
is called a sub sumption, the symbol =£[ the symbol of sub- 
sumption. 

E. PEANO 

The term of this whole development is to be found in the sym- 
bolism which Giuseppe Peano published in 1889. This comprises 
essentially more than the Boolean calculus and at the same time 
brings the latter to its final form. Its essentials will be given below 
(41.20). 



306 



III. PROPOSITIONAL LOGIC 



§41. PROPOSITIONAL LOGIC: BASIC CONCEPTS 
AND SYMBOLISM 

We speak first of the development of proposition-determining 

functors and other fundamental parts of proposition;! I Logic. This 
was first formulated, in the modern period of Logic, by Boole - 
actually as the second possible interpretation of his calculus (1847 . 
A more exact exposition appears in McColl (1877). Frege's Begrif/s- 
schrift (1879) marks a new beginning, in this as in so many other 
regions of formal logic. In connection with Frege's doctrine of 
implication we give also two important texts from Peirce. 

Later, Peano (1889) introduced a symbolism which is notably 
easier to read than Frege's; Russell's displays only inessential 
variations from it. But the symbolism which Lukasiewicz later 
constructed, in dependence on Frege, is basically different from 
Peano's. 

A. BOOLE 
We read in the Analysis: 

41.01 Of the conditional syllogism there are two, and only 
two formulae. 
1st The constructive, 

If A is B, then C is D, 

But A is B, therefore C is D. 
2nd The destructive, 

If A is B, then C is D, 

But C is not D, therefore A is not B. 
... If we examine either of the forms of conditional syllog- 
ism above given, we shall see that the validity of the argument 
does not depend upon any considerations which have reference 
to the terms A, B, C, D, considered as the representatives of 
individuals or of classes. We may, in fact, represent the 
Propositions A is B, C is D, by the arbitrary symbols X and Y 
respectively, and express our syllogisms in such forms as the 
following : 

If X is true, then Y is true, 

But X is true, therefore Y is true. 
Thus, what we have to consider is not objects and classes 
of objects, but the truths of Propositions, namely, of those 
elementary Propositions which are embodied in the terms of 
our hypothetical premises. 

307 



MATHEMATICAL VARIETY OF LOGIC 

41.02 If we confine ourselves to the contemplation of a 
given proposition X, and hold in abeyance every other conside- 
ration, then two cases only are conceivable, viz. first that the 
given Proposition is true, and secondly that it is false. As 
these cases together make up the Universe of the Proposition, 
and as the former is determined by the elective symbol x, 
the latter is determined by the symbol 1 - x. 

But if other considerations are admitted, each of these 
cases will be resolvable into others, individually less extensive, 
the number of which will depend upon the number of foreign 
considerations admitted. Thus if we associate the Propositions 
X and y, the total number of conceivable cases will be found 
as exhibited in the following scheme. 

Cases Elective expressions 

1st X true, y true xy 

2nd X true, Y false x (1 -y) 

3rd X false, Y true (1 -x) y 

4th X false, Y false (l-x){l-y). 

41.03 And it is to be noted that however few or many those 
circumstances may be, the sum of the elective expressions 
representing every conceivable case will be unity. Thus let 
us consider the three Propositions. X, It rains, Y, It hails, 
S, It freezes. The possible cases are the following: 

Cases Elective expressions 

1st It rains, hails, and freezes, xyz 

2nd It rains and hails, but does not 

freeze xy (1 - z) 

3rd It rains and freezes, but does not 

hail xz (1 - y) 

4th It freezes and hails, but does not yz (1 - x) 

rain 
5th It rains, but neither hails nor x (1 - y) (1 - z) 

freezes 
6th It hails, but neither rains nor 

freezes y (1 -x) (1 - z) 

7th It freezes, but neither hails nor z (1 - x) (1 - y) 

rains 
8th It neither rains, hails, nor freezes (1 - x) (1 - y) (1 - z) 

1 — sum 



308 



PROPOSITIONAL LOGIC 

41.04 ... To express that a given Proposition X is true. 

The symbol 1 - x selects those cases in which the Propo- 
sition X is false. But if the Proposition is true, there are no 
such cases in its hypothetical Universe, therefore 

1 - x = 0, 
or x = 1. 

To express that a given Proposition X is false. 

The elective symbol x selects all those cases in which the 
Proposition is true, and therefore if the Proposition is false, 

x = 6. 

These principles are then applied just like those of the logic of 
classes, to syllogistic practice. 

The similarity of the table of four cases in 41.02 with Philo's 
matrix of truth-values (20.07) is to be noticed. As has already been 
said, the Boolean calculus had no symbol for implication, nor yet 
one for negation; both are introduced by means of more complex 
formulae. In place of the logical sum, Boole had the notion of 
exclusive disjunction. Hence it is that propositional logic is made 
to appear as a discipline co-ordinate with, if not subordinate to, 
the logic of classes, by contrast to the clear insight possessed by the 
Stoics and Scholastics into its nature as basic. 

B. McCOLL 

Passing over the development that occurred between 1847 and 
1877, mainly due to Jevons and Peirce, we now give instead a text 
from Hugh McGoll (1877) in which propositional logic is emancipated 
from the calculus of classes, and endowed with all the symbols just 
mentioned. In a way this text marks the highest level of mathe- 
matical logic before Frege. 

41.05 Definition 1. - Let any symbols, say A, B, C, etc., 
denote statements [or propositions] registered for con- 
venience of reference in a table. Then the equation A = 1 
asserts that the statement A is true; the equation A = 
asserts that the statement A is false ; and the equation A = B 
asserts that A and B are equivalent statements. 

41.06 Definition 2. - The symbol A x B x C or ABC 
denotes a compound statement, of which the statements 
A, B, C may be called the factors. The equation ABC = 1 
asserts that all the three statements are true; the equation 
ABC = asserts that all the three statements are not true, 
i.e. that at least one of the three is false. Similarly a com- 
pound statement of any number of factors may be defined. 

309 



MATHEMATICAL VARIETY OF LOGIC 

41.07 Definition 3. - The symbol A + B + C denotes an 
indeterminate statement, of which the statements A, B, C 
may be called the terms. The equation A + B + C = asserts 
that all the three statements are false ; the equation A + B + C 
= 1 asserts that all the three statements are not false, i.e., 
that at least one of the three is true. Similarly an indeterminate 
statement of any number of terms may be defined. 

41.08 Definition 4. - The symbol A' is the denial of the 
statement A. The two statements A and A' are so related 
that they satisfy the two equations A + A' = 1 and A A ' = 0; 
that is to say, one of the two statements (either A or A') must 
be true and the other false. The same symbol {i.e. a dash) 
will convert any complex statement into its denial. For 
example, (AB)' is the denial of the compound statement 
AB. . . . 

41.09 Definition 5. - When only one of the terms of an 
indeterminate statement A + B + C + . . . can be true, or 
when no two terms can be true at the same time, the terms 
are said to be mutually inconsistent or mutually exclusive. 

41.10 Definition 12. - The symbol A: B [which may be call- 
ed an implication] asserts that the statement A implies B; or 
that whenever A is true B is also true. 

Note. - It is evident that the implication A:B and the 
equation A = AB are equivalent statements. 

C. FREGE 
1. Content and judgment 

A new period of propositional logic begins with Gottlob Frege. 
His first work, the Begriffsschrift of 1879, already contains in brief 
an unusually clear and thorough presentation of a long series of 
intuitions unknown to his immediate predecessors, while those 
already familiar are better formulated. To start with we choose a 
text relevant rather to semantics than logic, in which this great 
thinker introduces of his propositional logic with the 'judgment- 
stroke' : 

41.11 A judgment is always to be expressed by means of 
the sign 

This stands to the left of the sign or complex of signs in which 
the content of the judgment is given. If we omit the little 
vertical stroke at the left end of the horizontal stroke, then 

310 



PROPOSITIONAL LOGIC 

the judgment is to be transformed into a mere complex of 
ideas; the author is not expressing his recognition or non- 
recognition of the truth of this. Thus, let 

I— A 
mean the judgment: 'unlike magnetic poles attract one 
another'. In that case 

A 

will not express this judgment; it will be intended just to 
produce in the reader the idea of the mutual attraction of 
unlike magnetic poles - so that, e.g., he may make inferences 
from this thought and test its correctness on the basis of 
these. In this case we qualify the expression with the words 
'the circumstance thaf or 'the proposition lhal\ 

Not every content can be turned into a judgment by 
prefixing | — to a symbol for the content; e.g., the idea 'house' 
cannot. Hence we distinguish contents that are, and contents 
that are not, possible contents of judgment. 

As a constituent of the sign | — the horizontal stroke combines 
the symbols following it into a whole; assertion, which is expressed 
by the vertical stroke at the left end of the horizontal one, relates 
to the whole thus formed. The horizontal stroke I wish to call 
the content-stroke, and the vertical the judgment-stroke. The 
content-stroke is also to serve the purpose of relating any sign 
whatsoever to the whole formed by the symbols following 
the stroke. The content of what follows the content-stroke must 
always be a possible content of judgment. 

2. Implication 

Frege then introduces the Philonian concept of implication, 
though, unlike Peirce (41.14) he knows nothing in this connection 
of Philo or the Scholastics. It is remarkable that he proceeds almost 
exactly like Philo. 

41.12 If A and B stand for possible contents of judgment, 
we have the four following possibilities : 
(i) A affirmed, B affirmed; 
(ii) A affirmed, B denied; 
(iii) A denied, B affirmed; 
(iv) A denied, B denied. 

l — B 

stands for the judgment that the third possibility is not 

311 



MATHEMATICAL VARIETY OF LOGIC 

realized, but one of the other three is. Accordingly, the denial of 

l — B 
is an assertion that the third possibility is realized, i.e. that 
A is denied and B affirmed. 
From among the cases where 

•— B 

is affirmed, the following may be specially emphasized: 

(1) A is to be affirmed. - In this case the content of B is 
quite indifferent. Thus, let i — A mean: 3 x 7 = 31 ; let £ 
stand for the circumstance of the sun's shining. Here only 
the first two cases out of the four mentioned above are 
possible. A causal connection need not exist between the two 
contents. 

(2) B is to be denied. - In this case the content of A is 
indifferent. E.g., let B stand for the circumstance of perpetual 
motion's being possible, and A for the circumstance of the 
world's being infinite. Here only the second and fourth of the 
four cases are possible. A causal connection between A and B 
need not exist. 

(3) One may form the judgment 

l — B 
without knowing whether A and B are to be affirmed or denied. 
E.g., let B stand for the circumstance of the Moon's being in 
quadrature with the Sun, and A the circumstance of her 
appearing semi-circular. In this case we may render 

l — B 

by means of the conjunction 'if; 'if the Moon is in quadrature 
with the Sun, then she appears semi-circular'. The causal 
connection implicit in the word 'if is, however, not expressed 
by our symbolism; although a judgment of this sort can be 
made only on the ground of such a connection. For this con- 
nection is something general, and as yet we have no expression 
for generality. 

The text needs some explanations. First, Frege uses A 1 for the 
consequent and ' B' for the antecedent - contrary to ordinary usage, 
but like Aristotle; so the antecedent stands in the lower place. So 

312 



PROPOSITIONAL LOGIC 

the schema excludes only the case where the antecedent (B) is true 
and the consequent (A) is false; in all the other three cases the 
proposition is true. Thus we have just the same state of affairs as in 
Philo (20.07): the schema is a symbol of Philonian implication. It 
signifies 'if B, then A 1 in the Philonian sense of 'if. 

Important is the stress laid on the fact that implication has 
nothing to do with the causal connection between the facts signified 
by the antecedent and consequent. . 



D. PEIRCE 

Philonian implication alone continued to be used in mathematical 
logic up to 1918 - unlike usage in the Stoic and Scholastic periods. 
One of the best justifications of this concept which seems so odd to 
the man in the street, is to be found in a fairly late text of Peirce's, 
dated 1902. 

41.13 To make the matter clear, it will be well to begin by 
defining the meaning of a hypothetical proposition, in general. 
What the usages of language may be does not concern us; 
language has its meaning modified in technical logical for- 
mulae as in other special kinds of discourse. The question 
is what is the sense which is most usefully attached to the 
hypothetical proposition in logic ? Now, the peculiarity of the 
hypothetical proposition is that it goes out beyond the actual 
state of things and declares what would happen were things 
other than they are or may be. The utility of this is that 
it puts us in possession of a rule, say that 'if A is true, B is 
true', such that should we hereafter learn something of which 
we are now ignorant, namely that A is true, then by virtue 
of this rule, we shall find that we know something else, namely, 
that B is true. There can be no doubt that the Possible, in its 
primary meaning, is that which may be true for aught we 
know, that whose falsity we do not know. The purpose is 
subserved, then, if throughout the w r hole range of possibility, 
in every state of things in which A is true, B is true too. 
The hypothetical proposition may therefore be falsified by a 
single state of things, but only by one in which A is true 
while B is false. States of things in which A is false, as well 
as those in which B is true, cannot falsify it. If, then, B is a 
proposition true in every case throughout the whole range of 
possibility, the hypothetical proposition, taken in its logical 
sense, ought to be regarded as true, whatever may be the 

313 



MATHEMATICAL VARIETY OF LOGIC 

usage of ordinary speech. If, on the other hand, A is in no 
case true, throughout the range of possibility, it is a matter 
of indifference whether the hypothetical be understood to 
be true or not, since it is useless. But it will be more simple 
to class it among true propositions, because the cases in which 
the antecedent is false do not, in any other case, falsify a 
hypothetical. This, at any rate, is the meaning which I shall 
attach to the hypothetical proposition in general, in this 
paper. 

Also of interest is the following remark of the same logician 

(1896): 

41.14 Although the Philonian views lead to such incon- 
veniences as that it is true, as a consequence de inesse, that if 
the Devil were elected president of the United States, it 
would prove highly conducive to the spiritual welfare of the 
people (because he will not be elected), yet both Professor 
Schroder and I prefer to build the algebra of relatives upon 
this conception of the conditional proposition. The incon- 
venience, after all, ceases to seem important, when we reflect 
that, no matter what the conditional proposition be under- 
stood to mean, it can always be expressed by a complexus of 
Philonian conditionals and denials of conditionals. 



E. APPLICATIONS OF HIS SYMBOLISM BY FREGE 

Some examples of the applications of Frege's implication-schema 
will make his main ideas clearer. 

41.15 The vertical stroke joining the two horizontal ones 
is to be called the conditional-stroke. . . . Hence it is easy to 
see that 

I r A 

] — B 

— r 



denies the case in which A is denied, B and V are affirmed. 
This must be thought of as compounded of 

,4 and/ 1 

just as 
314 



PROPOSITIONAL LOGIC 



"TZb 

is from A and B. Thus we first have the denial of the case 
in which 

is denied, Y is affirmed. But the denial of 

signifies that A is denied and B is affirmed. Thus we obtain 
what is given above. 

41.16 From the explanation given in § 5 (41.12) it is obvious 
that from the two judgments 

| — , — A and I — B 
\—B 

there follows the new judgment i — A. Of the four cases 
enumerated above, the third is excluded by 

\—B 

and the second and fourth by : 

\— B, 
so that only the first remains. 

41.17 Let now X for example signify the judgment 

I— B 

— or one which j — , — A contains as a particular case. Then 
I— B 

I write the inference thus: 

\— B 

(X): 

I— A. 
Here it is left to the reader to put together the judgment 

] — B 

from | — B and | — A, and see that it tallies with the cited 
judgment X. 

315 



MATHEMATICAL VARIETY OF LOGIC 
F. NEGATION AND SUM IN FREGE 

Frege uses the same schemata, together with the 'negation-stroke', 
to express the logical sum. 

41.18 If a small vertical stroke is attached to the lower 
side of the content-stroke, this shall express the circumstance 
of the content's not being the case. Thus, e.g., the meaning 
of 

h-r- A: 
is: 'A is not the case'. I call this small vertical stroke the 
negation-stroke. 

41.19 We now deal with some cases where the symbols 
of conditionality and negation are combined. 

\—B 

means : 'the case in which B is to be affirmed and the negation 
of A is to be denied does not occur'; in other words, 'the 
possibility of affirming both A and B does not exist', or A and 
B are mutually exclusive'. Thus only the three following 
cases remain: 

A affirmed, B denied; 

A denied, B affirmed; 

A denied, B denied. 
From what has already been said, it is easy to determine 
the meaning possessed by each of the three parts of the 
horizontal stroke preceding A. 

means: 'the case in which A is denied and negation of B is 
affirmed does not exist'; or, l A and B cannot both be denied'. 
There remain only the following possibilities : 

A affirmed, B affirmed; 

A affirmed, B denied; 

A denied, B affirmed. 
A and B between them exhaust all possibilities. Now the 
words 'or', 'either - or', are used in two ways. In its first 
meaning, 

means just the same as 
316 



A or B' 



A 
B, 



PROPOSITIONAL LOGIC 

i. e. that nothing besides A and B is thinkable. E.g., if a 
gaseous mass is heated, then either its volume or its pressure 
increases. Secondly, the expression 

l A or B' 
may combine the meaning of 

1 F" A and that of [ ~T~ A 
— B 4" B 

so that (i) there is no third possibility besides A and B, (ii) A 

and B are mutually exclusive. In that case only the following 

two possibilities remain out of the four: 

A affirmed, B denied; 

A denied, B affirmed. 
Of these two uses of the expression 'A or J3' the more impor- 
tant is the first, which does not exclude the coexistence of A 
and B; and we shall use the word l or' with this meaning. Perhaps 
it is suitable to distinguish between 'or' and 'either - or', 
regarding only the latter as having the subsidiary meaning 
of mutual exclusion. 

G.PEANO'S SYMBOLISM FOR PROPOSITIONAL LOGIC 

Frege's symbolism has the unusual feature of being two-dimen- 
sional. In that it diverges from the historical practice of mankind 
which has almost always expressed its thoughts in one-dimensional 
writing. It must be admitted that this revolutionary novelty has 
much to be said for it - it notably widens the expressive possibilities 
of writing. But this was too revolutionary; Frege's symbolism did 
not prove generally intelligible, and the subsequent development took 
place in another direction. Schroder made no reference to it in 1892, 
Russell admitted in 1903 that he had learned much from Frege when 
he had met his system, but not having known it he followed Peano. 
Modern mathematical logic, though its authors have less depth of 
thought than Frege, has adopted Peano's symbolism. For this 
reason we quote a text from Peano's Arithmetices Principia (1889) in 
which he lays down this intuitively clear and meaningful symbolism 
for propositional logic. 

41.20 I. Concerning punctuation 

By the letters a, b, ... x, y, ... x', y' ... we indicate any 
undetermined beings. Determined beings we indicate by the 
signs or letters P, K, N. . . . 

For the most part we shall write signs on one and the same 
line. To make clear the order in which they are to be con- 
joined we use parentheses as in algebra, or dots .:/.:: etc. 

317 



MATHEMATICAL VARIETY OF LOGIC 

That a formula divided by dots may be understood, first 
the signs which are separated by no dots are to be collected, 
afterwards those separated by one dot, then those by two 
dots, etc. 

E.g. let a, 6, c, . . . be any signs. Then ab • cd signifies (ab)(cd) ; 
and ab . cd : ef . gh .-. k signifies (((ab) (cd)) ((ef) (gh))) k. 

Signs of punctuation may be omitted if there are formulae 
with different punctuation but the same sense; or if only 
one formula, and that the one we wish to write, has the 
sense. 

To avoid danger of ambiguity we never make use of . : as 
signs of arithmetical operations. 

The only form of parentheses is (). If dots and paren- 
theses occur in the same formula, signs contained in paren- 
theses are to be collected first. 

//. Concerning propositions 

By the sign P is signified a proposition. 

The sign r» is read and. If a, b are propositions; then a r» b 
is the simultaneous affirmation of the propositions a, b. For 
the sake of brevity we shall commonly write ab in place 
of a n b. 

The sign - is read not. Let abeaP; then -a is the negation 
of the proposition a. 

The sign w is read or (vet). Let a, b be propositions; then 
a v b is the same as - \-a.-b. 

[By the sign V is signified verum or identity; but we never 
use this sign.] 

The sign A signifies falsum or absurdum. 

[The sign C signifies is a consequence; thus bCa is read b 
is a consequence of the proposition a. But we never use this 
sign.] 

The sign j signifies is deduced (deducitur) : thus a j b signi- 
fies the same as bCa. 



H. LATER DEVELOPMENT OF SYMBOLISM 
FOR PROPOSITIONAL LOGIC 

Peano's successors introduced only minor alterations to his 
symbolism. First Russell (1903), who writes 'v' instead of 'w', and 
l ~ instead of '-', then Hilbert and Ackermann (1928) who write 
a stroke over a letter for negation, and 'oo' for the equivalence- 

318 



PROPOSITIONAL LOGIC 

sign ' = ' of Frege and Russell. M. H. ShefTer (1928) introduced '|' as a 
sign for 'not both'. 

The Polish school, on the other hand, developed two symbolic 
languages essentially different from Peano's; those of St. Lesniewski 
and J. Lukasiewicz. We shall not go into the first, which is peculiar 
and little used, but the symbolism of Lukasiewicz deserves brief 
exposition, both for its originality and its exactness. The essential 
feature is that all predicates (called by Lukasiewicz 'functors'; 
stand in front of their arguments; thus all brackets and dots are 
dispensed with, without any ambiguity arising. 

The various sets of symbols may be compared thus: 
McColl A' + x : ' = 



Peano 


-P 


w 


r\ 


D 


= 


Russell 


r+~>p 


V 


. 


D 


= 


Hilbert 


A 




& 


-> 


CO 


Lukasiewicz 


Np 


A 


K 


c 


E 



D [i.e.NK) 

Thus Lukasiewicz writes 'Cpq' for 'p d q\ and l Apq' for 'p v q' . 
An example of a more complex formula is 'CCpqCNqNp' instead 
of 'p D q . D . ^ q D r*** p\ 



§42. FUNCTION, VARIABLE, TRUTH-VALUE 

While nearly everything mentioned so far is within the scope of 
Stoic and Scholastic logic - though the new logicians knew hardly 
anything of the achievements of their predecessors, - the concepts of 
function, variable, and truth-value, without effecting anything radi- 
cally new, yet produce so marked a development of the old concept 
of logical form as to deserve distinct and thorough treatment. 

After an introductory quotation from De Morgan (1858) con- 
cerning logical form in general, we give Frege's fundamental text on 
the concept of function (1893), the explanation and development of 
Frege's thought from Russell and the Principia (1903 and 1910), 
and finally the extension of the concept of function to many-place 
functions by Peirce (1892) and Frege (1893). For the doctrine of 
the variable in mathematical logic we have a quotation from Frege's 
Begriffsschrift (1879) and Russell's elaboration of the ideas therein 
(1903 and 1910). As to truth-values, two texts from Frege (1893) 
and one from Peirce (1885) are to hand. 

In conclusion we exhibit some examples of modern truth-matrices 
(truth-tables, tables of truth-values), by which propositional func- 
tors are defined, taking these from Peirce (1902) and Wittgenstein 
(1921); the decision procedure based on them is illustrated from 
Kotarbinski (1929). 



319 



MATHEMATICAL VARIETY OF LOGIC 

A. LOGICAL FORM 

An important text of De Morgan's make a fitting start. It dates 
from 1858, and shows a very clear idea of logical form. It may be 
compared with Buridan's definition of logical form (26.12): the 
thought is the same, but more developed, in that abstraction is made 
from the sense of the logical constants. 

42.01 In the following chain of propositions, there is 
exclusion of matter, form being preserved, at every step: - 

Hypothesis 
(Positively true) Every man is animal 

Every man is Y Y has existence 

Every X is Y X has existence 

Every X Y is a transitive 

relation 

a of X Y a a fraction < or = 1 

(Probability p) (3 of X Y pa fraction < or = 1 

The last is nearly the purely formal judgment, with not a 
single material point about it, except the transitiveness of the 

copula. But 'is' is more intense than the symbol , which 

means only transitive copula: for 'is' has transitiveness, and 
more. Strike out the word transitive, and the last line shews the 
pure form of the judgment. 

The foregoing table is to be understood in the sense that the 
conditions formulated in one row, hold for all subsequent rows; thus, 

e.g., the relation shown by the stroke (' ') in the two last rows 

must be transitive, since this is laid down in the preceding row. 

Neither De Morgan nor any other logician can remain at so high 
a level of abstraction as is here achieved. Basically, this is a re- 
discovery of the scholastic concept of form, made through a broaden- 
ing of the mathematical concept of function, for which we refer to 
Peirce (42.02) and Frege. 

B. CONCEPT OF FUNCTION: FREGE 

We now give Frege's fundamental text (1893): 

42.03 If we are asked to state what the word 'function' as 
used in mathematics originally stood for, we easily fall into 
saying that a function of x is an expression formed, by means 
of the notations for sum, product, power, difference, and so 
on, out of V and definite numbers. This attempt at a 
definition is not successful because a function is here said to 

320 



PROPOSITIONAL LOGIC 

be an expression, a combination of signs, and not what the 
combination designates. Accordingly another attempt would 
be made: we could try 'reference of an expression' instead of 
'expression'. There now appears the letter V which indicates 
a number, not as the sign '2' does, but indefinitely. For 
different numerals which we put in the place of V, we get, in 
general, a different reference. Suppose, e.g., that in the 
expression '(2 + 3 • x 2 ) x\ instead of '#' we put the number- 
signs '0', '1', '2', '3', one after the other; we then get corres- 
pondingly as the reference of the expression the numbers 
0, 5, 28, 87. Not one of the numbers so referred to can claim 
to be our function. The essence of the function comes out 
rather in the correspondence established between the numbers 
whose signs we put for V and the numbers which then appear 
as the reference of our expression - a correspondence which is 
represented intuitively by the course of the curve whose 
equation is, in rectangular co-ordinates, 'y = (2 + 3. x 2 ) x\ In 
general, then, the essence of the function lies in the part of 
the expression which is there over and above the l x\ The 
expression of a function needs completion, is 'unsaturated' . The 
letter l x' only serves to keep places open for a numerical sign 
to be put in and complete the expression; and thus it enables 
us to recognize the special kind of need for a completion that 
constitutes the peculiar nature of the function symbolized 
above. In what follows, the Greek letter %' will be used 
instead of the letter V. This 'keeping open' is to be understood 
in this way: All places in which '£' stands must always be 
filled by the same sign and never by different ones. I call these 
places argument-places, and that whose sign or name takes 
these places in a given case I call the argument of the function 
for this case. The function is completed by the argument: 
I call what it becomes on completion the value of the function 
for the argument. We thus get a name of the value of a func- 
tion for an argument when we fill the argument-places in the 
name of the function with the name of the argument. Thus, 
e.g., '(2 + 3 . I 2 ) 1' is name of the number 5, composed of the 
function-name '(2 + 3 . £ 2 ) £' and '1'. The argument is not to 
be reckoned in with the function, but serves to complete the 
function, which is 'unsaturated' by itself. When in the sequel 
an expression like 'the function O (£)' is used, it is always to be 
observed that the only service rendered by '£' in the symbol 
for the function is that it makes the argument-places recogniz- 

321 



MATHEMATICAL VARIETY OF LOGIC 

able; it does not imply that the essence of the function 
becomes changed when any other sign is substituted for '£'. 

The following remarks will assist understanding of this pioneer 
passage. In mathematical usage the word 'function' has two refer- 
ences, usually not very clearly distinguished. On the one hand it 
stands for an expression (formula) in which a variable occurs, on the 
other for the 'correspondence between numbers' for which such an 
expression stands, and so for some kind of lecton or in general, for 
that for which the expression stands, (which in any case is not a 
written symbol). Frege makes a sharp distinction between these two 
references, and allows only the second to the word 'function' - 
conformably with his general position that logic (and mathematics) 
has as its object not symbols but what they stand for. It is important 
to understand this, because Russell and nearly all logicians after him 
will speak of expressions and formulae as 'functions', unlike Frege. 

However, this opposition is irrelevant to the basic logical problems 
considered here. Frege, too, makes use of analysis of expressions to 
convey his thought, and what he states in the text just quoted, 
holds good for every interpretation of the word 'function'. He intro- 
duces, namely, three fundamental concepts: 1. of the argument and 
argument-place, 2. of a value, 3. of an 'unsaturated' function, i.e. 
one containing a variable. 

C. PROPOSITIONAL FUNCTION: RUSSELL 

Russell who knew the work of Frege well, followed his ideas but 
with some divergences. He seems to start from the Aristotelian 
concept of proposition rather than from the mathematical concept 
of function, and as already said, apparently interprets the word 
'function' as the name of an expression or written formula. In the 
Principles (1903) he writes: 

42.04 It has always been customary to divide propositions 
into subject and predicate; but this division has the defect 
of omitting the verb. It is true that a graceful concession is 
sometimes made by loose talk about the copula, but the verb 
deserves far more respect than is thus paid to it. We may say, 
broadly, that every proposition may be divided, some in 
only one way, some in several ways, into a term (the subject) 
and something which is said about the subject, which some- 
thing I shall call the assertion. Thus 'Socrates is a man' may 
be divided into Socrates and is a man. The verb, which is the 
distinguishing mark of propositions, remains with the asser- 
tion; but the assertion itself, being robbed of its subject, is 
neither true nor false. . . . 

322 



PROPOSITIONAL LOGIC 

If this text is compared with 12.01 and similar passages in Aris- 
totle, it can be seen that Russell here opts for the original Aristotelian 

analysis of propositions against that of the later 'classical' logic. 
In this connection he seems to have been the first to formulate 
expressly the idea that when the subject is replaced by a variable, the 
resulting formula - the propositional function is no longer a 
proposition. The same problem is still more explicitly treated in the 
Principia: 

42.05 By a 'propositional function' we mean something 
which contains a variable x, and expresses a proposition as 
soon as a value is assigned to x. That is to say, it differs from 
a proposition solely by the fact that it is ambiguous: it 
contains a variable of which the value is unassigned. . . . 

42.06 The question as to the nature of a function is by no 
means an easy one. It would seem, however, that the essential 
characteristic of a function is ambiguity. Take, for example, 
the law of identity in the form l A is A\ which is the form in 
which it is usually enunciated. It is plain that, regarded 
psychologically, we have here a single judgment. But what 
are we to say of the object of judgment? We are not judging 
that Socrates is Socrates, nor that Plato is Plato, nor any 
other of the definite judgments that are instances of the law 
of identity. Yet each of these judgments is, in a sense, within 
the scope of our judgment. We are in fact judging an ambi- 
guous instance of the propositional function 'A is A'. We appear 
to have a single thought which does not have a definite 
object, but has as its object an undetermined one of the values 
of the function 'A is A'. It is this kind of ambiguity that 
constitutes the essence of a function. When we speak of l yx', 
where x is not specified, we mean one value of the function, 
but not a definite one. We may express this by saying that 'yx' 
ambiguously denotes cpa, 96, cpc, etc., where cpa, 96, 9c, etc. 
are the various values of l yx'. 

D. MANY-PLACE FUNCTIONS 

Perhaps even more important than the broadening of the concept 
of function to include non-mathematical domains, is the extension to 
many-place functions achieved by Frege and Peirce. The resulting 
extension of the Aristotelian subject-predicate schema is something 
quite new in formal logic. Our first text is Peirce's (1892) : 

42.07 If upon a diagram we mark two or more points to 
be identified at some future time with objects in nature, so as 

323 



MATHEMATICAL VARIETY OF LOGIC 

to give the diagram at that future time its meaning; or if in 
any written statement we put dashes in place of two or more 
demonstratives or pro-demonstratives, the professedly incom- 
plete representation resulting may be termed a relative rhema. 
It differs from a relative term only in retaining the 'copula', or 
signal of assertion. If only one demonstrative or pro-demon- 
strative is erased, the result is a non-relative rhema. For 

example, ' buys from for the price of ', is a 

relative rhema; it differs in a merely secondary way from 

' is bought by from for ' , 

from ' sells --- to for ', 

and from ' is paid by to for '. 

On the other hand, ' is mortal' is a non-relative rhema. 

42.08 A rhema is somewhat closely analogous to a chemical 
atom or radicle with unsaturated bonds. A non-relative 
rhema is like a univalent radicle; it has but one unsaturated 
bond. A relative rhema is like a multivalent radicle. The 
blanks of a rhema can only be filled by terms, or, what is the 
same thing, by 'something which' (or the like) followed by a 
rhema; or, two can be filled together by means of 'itself or 
the like. So, in chemistry, unsaturated bonds can only be 
saturated by joining two of them, which will usually, though 
not necessarily, belong to different radicles. If two univalent 
radicles are united, the result is a saturated compound. So, 
two non-relative rhemas being joined give a complete propo- 
sition. Thus, to join ' is mortal' and 'is a man', we 

have 'A is mortal and AT is a man', or some man is mortal. So 
likewise, a saturated compound may result from joining two 
bonds of a bivalent radicle; and, in the same way, the two 
blanks of a dual rhema may be joined to make a complete 

proposition. Thus, ' loves ', 'A loves A', or something 

loves itself. 

Frege, a year later, writes in the same sense: 

42.09 So far we have only spoken of functions of one 
argument; but we can easily make the transition to func- 
tions with two arguments. These need a double completion in 
that after a completion by one argument has been effected, a 
function with one argument is obtained. Only after another 
completion do we reach an object, which is then called the 
value of the function for the two arguments. Just as we made 

324 



PROPOSITIONAL LOGIC 

use of the letter '£' for functions with one argument, so we now 
use the letters '£' and %' to express the twofold unsaturatedness 
of functions with two arguments, as in '(<; + £) 2 + £\ In 
substituting, e.g., '1' for '£' we saturate the function to the 
extent that in (§ + l) 2 + 1 we are left with a function with 
only one argument. This way of using the letters '£' and '£' 
must always be kept in view when an expression such as 'the 
function T (£, £)' occurs (cf. 42.03). ... I call the places in 
which '£' appears, ^-argument-places, and those in which '£' 
appears, ^-argument- places. I say that the ^-argument- 
places are mutually cognate and similarly the ^-argument- 
places, while I call a ^-argument-place not cognate to a £- 
argument-place. 

The functions with two arguments \ = £ and \ > £ always 
have a truth-value as value [at least when the signs '=' 
and '>' are appropriately explained]. For our purposes we 
shall call such functions relations. E.g., 1 stands to 1 in the 
first relation, and generally every object to itself, while 2 
stands to 1 in the second relation. We say that the object T 
stands to the object A in the relation T (£, ?) if Y (I\ A) is the 
True. Similarly we say that the object A falls under the con- 
cept O (£), if O (A) is the True. It is naturally presupposed 
here, that the function O (£) always has a truth-value. (Foot- 
note of Frege' s : A difficulty occurs which can easily obscure the 
true state of affairs and so cast doubt on the correctness of 
my conception. When we compare the expression 'the truth- 
value of this, that A falls under the concept O (£)' with 'O 
(A)', we see that to '® ( )' there properly corresponds 'the 
truth-value of this, that ( ) falls under the concept O (£)' and 
not 'the concept O (£)'. Thus the last words do not properly 
signify a concept [in our sense], though the form of speech 
makes it seem as if they do. As to the difficulty in which 
language thus finds itself, cf. my paper On Concept and Object.) 



E. THE VARIABLE 
1. Frege 

Variables, introduced by Aristotle, were subsequently regularly 
used both in logic and mathematics. A reflective concept of variable 
is already to be found in Alexander of Aphrodisias (24.08V In 
mathematical logic the concept of variable is first explicitly intro- 
duced by Frege. 

325 



MATHEMATICAL VARIETY OF LOGIC 

42.10 The symbols used in the general theory of magnitude 
fall into two kinds. The first consists of the letters ; each letter 
represents either an indeterminate number or an indeter- 
minate function. This indeterminateness makes it possible to 
express by means of letters the general validity of propositions ; 
e.g.: [a + b) c = ac + be. The other kind contains such 
symbols as +, — , V, 0, 1, 2; each of these has its own proper 
meaning. 

/ adopt this fundamental idea of distinguishing two kinds of 
symbols (which unfortunately is not strictly carried out in the 
theory of magnitude - footnote of Frege's: Consider the 
symbols 1, log, sin, Lim. -) in order to make it generally 
applicable in the wider domain of pure thought. Accordingly, 
I divide all the symbols I use into those that can be taken to 
mean various things and those that have a fully determinate 
sense. The first kind are letters, and their main task is to be 
the expression of generality. For all their indeterminateness, 
it must be laid down that a letter retains in a given context 
the meaning once given to it. 

2. Russell 

42.12 The idea of a variable, as it occurs in the present 
work, is more general than that which is explicitly used in 
ordinary mathematics. In ordinary mathematics, a variable 
generally stands for an undetermined number or quantity. In 
mathematical logic, any symbol whose meaning is not deter- 
minate is called a variable, and the various determinations of 
which its meaning is susceptible are called the values of the 
variable. The values may be any set of entities, propositions, 
functions, classes or relations, according to circumstances. 
If a statement is made about 'Mr A and Mr B\ 'Mr A' and 
'Mr B J are variables whose values are confined to men. A 
variable may either have a conventionally-assigned range of 
values, or may (in the absence of any indication of the range 
of values) have as the range of its values all determinations 
which render the statement in which it occurs significant. Thus 
when a text-book of logic asserts that l A is A', without any 
indication as to what A may be, what is meant is that any 
statement of the form 'A is A' is true. We may call a variable 
restricted when its values are confined to some only of those 
of which it is capable ; otherwise we shall call it unrestricted. 
Thus when an unrestricted variable occurs, it represents any 

326 



PROPOSITIONAL LOGIC 

object such that the statement concerned can be made 
significantly (i.e. either truly or falsely) concerning that 
object. For the purposes of logic, the unrestricted variable is 
more convenient than the restricted variable, and we shall 
always employ it. We shall find that the unrestricted variable 
is still subject to limitations imposed by the manner of its 
occurrence, i.e. things which can be said significantly con- 
cerning a proposition cannot be said significantly concerning 
a class or a relation, and so on. But the limitations to which 
the unrestricted variable is subject do not need to be explic- 
itly indicated, since they are the limits of significance of the 
statement in which the variable occurs, and are therefore 
intrinsically determined by this statement. This will be more 
fully explained later. 

To sum up, the three salient facts connected with the use 
of the variables are: (1) that a variable is ambiguous in its 
denotation and accordingly undefined; (2) that a variable 
preserves a recognizable identity in various occurrences 
throughout the same context, so that many variables can 
occur together in the same context each with its separate 
identity; and (3) that either the range of possible determina- 
tions of two variables may be the same, so that a possible 
determination of one variable is also a possible determination 
of the other, or the ranges of two variables may be different, 
so that, if a possible determination of one variable is given 
to the other, the resulting complete phrase is meaningless 
instead of becoming a complete unambiguous proposition 
(true or false) as would be the case if all variables in it had 
been given any suitable determinations. 

F. TRUTH-VALUES 

Truth-values, which are of great importance in formal logic, 
form a special kind of value. The idea is already present in the 
Megarian school (20.07), but its expression and first description 
comes from Frege. His doctrine is linked to his own semantics, 
according to which every proposition is a name for truth or falsity, 
and in this he has not been generally followed, but the concept of 
truth-value has been accepted by all. 

We give first a text of Frege's : 

42.13 But that indicates at the same time that the domain 
of values for functions cannot remain limited to numbers ; for if 

327 



MATHEMATICAL VARIETY OF LOGIC 

I take as arguments of the function £ 2 = 4 the numbers 0,1,2, 3, 
in succession, I do not get numbers. '0 2 = 4', '1 2 = 4', '2 2 = 4', 
'32 = 4' 5 are expressions now of true, now of false thoughts. 
I express this by saying that the value of the function £ 2 = 4 
is the truth-value either of what is true or of what is false. 
From this it can be seen that I do not intend to assert any- 
thing by merely writing down an equation, but that I only 
designate a truth-value; just as I do not intend to assert 
anything by simply writing down '2 2 ' but only designate 
a number. I say: The names '2 2 =4' and '3 >2' stand for the 
same truth-value' which I call for short the True. In the same 
manner '3 2 = 4' and '1 > 2' stand for the same truth-value, 
which I call for short the False, just as the name '2 2 ' stands for 
the number 4. Accordingly I say that the number 4 is the 
reference of '4' and of '2 2 ', and that the True is the reference 
of '3 > 2'. But I distinguish the sense of a name from its 
reference. The names '2 2 ' and ' 2+ 2' have not the same sense, 
nor have '2 2 = 4' and '2 + 2 = 4'. The sense of the name for a 
truth-value I call a thought. I say further that a name expresses 
is its sense, and what it stands for is its reference. I designate 
by a name that which it stands for. 

The function £ 2 = 4 can thus have only two values, the 
True for the arguments + 2 and - 2 and the False for all other 
arguments. 

Also the domain of what is admitted as argument must be 
extended - indeed, to objects quite generally. Objects stand 
opposed to functions. I therefore count as an object everything 
that is not a function: thus, examples of objects are numbers, 
truth-values, and the ranges to be introduced further on. The 
names of objects - or proper names - are not therefore accom- 
panied by argument-places, but are 'saturated', like the 
objects themselves. 

42.14 I use the words, 'the function 0(5) has the same 
range as the function Y(5)', to stand for the same thing as 
the words, 'the functions 0(5) and T(£) have the same value 
for the same arguments'. This is the case with the functions 
5 2 = 4 and 3.£ 2 = 12, at least if numbers are taken as argu- 
ments. But we can further imagine the signs of evolution and 
multiplication defined in such a manner that the function 
(5 2 = 4) = (3.£ 2 = 12) has the True as its value for any 
argument whatever. Here an expression of logic may also be 
used: 'The concept square root of 4 has the same extension as 

328 



PROPOSITIONAL LOGIC 

the concept something whose square when trebled makes 12\ With 
those functions whose value is always a truth-value we can 
therefore say 'extension of the concept' instead of 'range of the 
function', and it seems suitable to say that a concept is a 
function whose value is always a truth-value. 

Independently of Frege, Peirce developed similar thoughts in 
1885. His treatment of truth-values is more formalistic and not 
tied to any particular semantic theory. However, this formalism 
enabled him to formulate one which seems to qualify him to be 
regarded as a precursor of many-valued logics. 

42.15 According to ordinary logic, a proposition is either 
true or false, and no further distinction is recognized. This 
is the descriptive conception, as the geometers say; the metric 
conception would be that every proposition is more or less 
false, and that the question is one of amount. At present we 
adopt the former view. 

42.16 Let propositions be represented by quantities. Let 
v and f be two constant values, and let the value of the quan- 
tity representing a proposition be v if the proposition is true 
and be f if the proposition is false. Thus, x being a proposition, 
the fact that x is either true or false is written 

(x - f) (v - x) = 0. 
So 

(x - f) (v - g) = 
will mean that either x is false or y is true. . . . 

42.17 We are, thus, already in possession of a logical 
notation, capable of working syllogism. Thus, take the 
premisses, 'if x is true, y is true', and 'if y is true, z is true'. 
These are written 

(x - f) (v - g) = 
(y - f) (v - z) = 0. 
Multiply the first by (v - z) and the second by [x - f) and add. 
We get 

(x - I) (v - f) (v - z) = 0, 
or dividing by v - f, which cannot be 0, 
(x - f) (v - z) = 0; 
and this states the syllogistic conclusion, 'if x is true, z is 
true'. 

42.18 But this notation shows a blemish in that it express- 
es propositions in two distinct ways, in the form of quan- 
tities, and in the form of equations; and the quantities are 

329 



MATHEMATICAL VARIETY OF LOGIC 

of two kinds, namely those which must be either equal to 
f or to v, and those which are equated to zero. To remedy 
this, let us discard the use of equations, and perform no 
operations which can give rise to any values other than f 
and v. 

42.19 Of operations upon a simple variable, we shall need 
but one. For there are but two things that can be said about 
a single proposition, by itself; that it is true and that it is 
false, 

x = v and x = f. 
The first equation is expressed by x itself, the second by any 
function, 9, of x, fulfilling the conditions 
cpv = f <pf = v. 
This simplest solution of these equations is 
yx = i + v - x. 



G. TRUTH-MATRICES 
1. Peirce 

The standpoint revealed in the last text comes near to defining 
propositional functors by means of truth-values. Tabular definitions 
of this kind have already been met with in the Stoic-Megarian school 
(20.07 ff.), later on in Boole, though without explicit reference to 
truth-values, and finally in Frege's Begriffsschrift (41.12). Peirce has 
the notion quite explicitly, and in connection, moreover, with 
ancient logic, in 1880: 

42.20 There is a small theorem about multitude that it 
will be convenient to have stated, and the reader will do well 
to fix it in his memory correctly. ... If each of a set of m 
objects be connected with some one of a set of n objects, the 
possible modes of connection of the sets will number n m . 
Now an assertion concerning the value of a quantity either 
admits as possible or else excludes each of the values v and f. 
Thus, v and f form the set m objects each connected with one 
only of n objects, admission and exclusion. Hence there are, 
n m , or 2 2 , or 4, different possible assertions concerning the 
value of any quantity, x. Namely, one assertion will simply 
be a form of assertion without meaning, since it admits either 
value. It is represented by the letter, x. Another assertion 
will violate the hypothesis of dichotomies by excluding both 
values. It may be represented by x. Of the remaining two, 
one will admit v and exclude f, namely x; the other will 
admit f and exclude v, namely x. 

330 




PROPOSITIONAL LOGIC 

Now, let us consider assertions conce- 
rning the values of two quantities, xand 
y. Here there are two quantities, each 
of which has one only of two values; so 
that there are 2 2 , or 4, possible states of 
things, as shown in this diagram. 

Above the line, slanting upwards to 
the right, are placed the cases in which 
x is v; below it, those in which x is f. 
Above the line but slanting downward 

to the right, are placed the cases in which y is v; below it, 
those in which y is f. Now in each possible assertion each of 
these states of things is either admitted or excluded ; but not 
both. Thus m will be 2 2 , while n will be 2; and there will 
be n m , or 2 2 , or 16, possible assertions. . . . 

Of three quantities, there are 2 3 , or 8, possible sets of 
values, and consequently 2 8 , or 256, different forms of propo- 
sitions. Of these, there are only 38 which can fairly be said to 
be expressible by the signs [used in a logic of two quantities]. 
It is true that a majority of the others might be expressed 
by two or more propositions. But we have not, as yet, expressly 
adopted any sign for the operation of compounding propo- 
sitions. Besides, a good many propositions concerning three 
quantities cannot be expressed even so. Such, for example, 
is the statement which admits the following sets of values : 

x y z 



V 


V V 


V 


f f 


i 


V f 


i 


f V 



Moreover, if we were to introduce signs for expressing 
[each of] these, of which we should need 8, even allowing the 
composition of assertions, still 16 more would be needed to 
express all propositions concerning 4 quantities, 32 for 5, 
and so on, ad infinitum. 



2. Wittgenstein 

The same doctrine was systematically elaborated about 1920 by 
J. Lukasiewicz, E. L. Post and L. Wittgenstein. We give the relevant 
text from the last: 

42.21. With regard to the existence of n atomic facts there 

331 



MATHEMATICAL VARIETY OF LOGIC 



are 



K n = 2 ( ) possibilities. 



It is possible for all combinations of atomic facts to exist, 
and the others not to exist. 

42.22 To these combinations correspond the same number 
of possibilities of the truth - and falsehood - of n elementary 
propositions. 

42.23 The truth-possibilities of the elementary propositions 
mean the possibilities of the existence and non-existence of 
the atomic facts. 

42.24 The truth-possibilities can be presented by schemata 
of the following kind ('T' means 'true, l F' 'false'. The rows 
of T's and F's under the row of the elementary propositions 
mean their truth-possibilities in an easily intelligible sym- 
bolism). 



P q r 

T T T 

F T T 

~T~ F _T 

T T F 

F~ F ~f 

F T F 

T F F 

F F F 



p <L 

T T 

F T 

T F 
F F 



42.25 . . . The truth-possibilities of the elementary propo- 
sitions are the conditions of the truth and falsehood of the 
propositions. 

42.26 It seems probable even at first sight that the intro- 
duction of the elementary propositions is fundamental for 
the comprehension of the other kinds of propositions. Indeed 
the comprehension of the general propositions depends palpably 
on that of the elementary propositions. 

42.27 With regard to the agreement and disagreement of a 
proposition with the truth-possibilites of n elementary 

Kn (K \ 
propositions there are 2 ( n ) = L n possibilities. 

42.28 . . . Thus e.g. 



332 



PROPOSITIONAL LOGIC 



p 


q 




T 


T 


T 


F 


T 


T 


r 


F 




F 


F 


T 



is a prepositional sign. 

42.29 . . . Among the possible groups of truth-conditions 
there are two extreme cases. 

In the one case the proposition is true for all the truth- 
possibilities of the elementary propositions. We say that the 
truth-conditions are tautological. 

In the second case the proposition is false for all the truth- 
possibilities. The truth-conditions are self-contradictory. 

In the first case we call the proposition a tautology, in the 
second case a contradiction. 

42.30 The proposition shows what it says, the tautology 
and the contradiction that they say nothing. 

The tautology has no truth-conditions, for it is uncondi- 
tionally true; and the contradiction is on no condition true. 

Tautology and contradiction are without sense. 

(Like the point from which two arrows go out in opposite 
directions.) 

(I know, e.g. nothing about the weather, when I know 
that it rains or does not rain.) 

42.31 Tautology and contradiction are, however, not 
nonsensical; they are part of the symbolism, in the same 
way that '0' is part of the symbolism of Arithmetic. 

The name 'tautology' and the last quotation show the peculiar 
(extremely nominalist) tendency underlying Wittgenstein's semantic 
views. It is diametrically opposed to Frege's tendency and from his 
point of view misleading. 

H. DECISION PROCEDURE OF LUKASIEWICZ 

The tables of values constructed in the texts just cited provide a 
decision procedure for propositional functions, i.e. a procedure which 
enables one to decide whether a function is a logical law (whether it 
becomes a true proposition for every correct substitution). The basic 
idea of such a procedure is present in Schroder (42.32). It was 
developed by E.L. Post (42.33) and was known to J. Lukasiewicz at 
the same time (42.34). It is set out in full in the manuals of Hilbert 

333 



MATHEMATICAL VARIETY OF LOGIC 

and Ackermann (42.35), 1928, and T. Kotarbinski, 1929. We quote 
Kotarbinski's text, because of its clarity. The author writes 'p" for 
'not p', and uses '+', '<', ' = ', as signs of addition, implication and 
equivalence respectively. 

42.36 We shall now give a very simple method of verifica- 
tion for the propositional calculus, which enables one to 
verify the correctness of every formula in this domain [viz. 
the zero-one method of verification]. We stipulate for this 
purpose that it is permitted to write, say, zero for a false 
proposition, and one for a true. With the help of this sym- 
bolism we now investigate whether a given formula becomes 
a true proposition for all substitutions of propositions for 
propositional variables - always under the condition that the 
same (proposition) is substituted for the same (variable), or 
whether on the other hand it becomes a false proposition for 
some substitutions. In the first case it is a valid formula, in 
the second an invalid one. . . . 

We recollect in this connection: (1) that the negation of a 
true proposition is always a false proposition, and conver- 
sely; (2) that the logical product is true only when both 
factors are true; (3) that the logical sum is always true when 
at least one of its parts is true, and false only in the case 
that both its parts are false; (4) that implication is false 
only when its antecedent is true and its consequent false; 
(5) that equivalence is true only when both sides are true or 
both false; but when one side is true, the other false, then the 
whole equivalence is false. If, for example, we put zero for p 
and one for q in a formula, then this formula can be further 
simplified by writing a zero in place of the product of p and q 
wherever this occurs throughout the formula- and analogously 
in the case of the other functions. . . . 

If, after the application of all possible substitutions of 
zero and one for propositional variables in a given formula, 
and after carrying out the . . . simplifications described, the 
formula always reduces to a one, then it is true. But if it 
is zero for even a single choice of substitutions, it is invalid. 

To have an example, we verify the formula of transpo- 
sition . . . 

(p < q) = W < p') 

1. We suppose that true propositions have been substituted 
both for p and for q. Our formula then takes on the form 

(i < i) = (r < i'). 

334 



PROPOSITIONAL LOGIC 

Simplifying it by Rule (1), we get: 

(1<1) = (0<0) 
. . . further ... we get 

1 = 1 
which ... we can replace by 

1. 

The process is then repeated with the other three possible 
substitutions. 



§43. PROPOSITIONAL LOGIC AS A SYSTEM 

In the last two paragraphs we have spoken of the basic concepts 
and one of the main methods of modern propositional logic. Now we 
come on to show the second, axiomatic, method at work, in some 
sample sections reproduced from different propositional systems. 

These sections will be taken from the systems of McColl (1877), 
Frege (1879 and 1893), Whitehead and Russell (1910) - here we insert 
a text from Peirce, connected with the Sheffer stroke - and finally 
Lukasiewicz (1920). In this last system, two-valued propositional 
logic seems to have reached the term of its development. 

A. McCOLL 

The ensuing texts, which are a continuation of the definitions in 
41.05 ff., contain rules for an algebraic system of propositional logic, 
constructed in the spirit of Boole. It may be compared with the 
non-algebraic system of Lukasiewicz (43.45). It may be remarked 
that while this algebraic style has been for the most part superseded, 
there have also been quite recent algebraic systems (Tarski). 

43.01 Rule 1. - The rule of ordinary algebraical multi- 
plication applies to the multiplication of indeterminate 
statements, thus : 

A{B + C) =AB + AC; (A + B) (C + D) = AC + AD + BC + 

BD; 

and so on for any number of factors, and whatever be the 

number of terms in the respective factors. 

43.02 Rule 2. - Let A be any statement whatever, and let 
B be any statement which is implied in A [and which must 
therefore be true when A is true, and false when A is false]; 
or else let B be any statement which is admitted to be true 
independently of A; then [in either case] we have the equation 
A = AB. As particular cases of this we have A = AA = 

335 



MATHEMATICAL VARIETY OF LOGIC 

AAA = etc., as repetition neither strengthens nor weakens 
the logical value of a statement. Also, 
A =A{B+B')=A{B+B'){C+C')=etc., for 
B+B' = l=C+C'=etc. [see Def. 4] (41.08). 

43.03 Rule 3. - {ABy=AB , +A f B+A , B' 

=AB'+A , (B+B')=AB , +A' 
=A' B+B'(A+A')=A' B+B' , 
for A+A' — l and B + B' = 1. Similarly we may obtain various 
equivalents (with mutually inconsistent terms) for (ABC)', 
(ABCD)', etc. 

43.04 Rule 4. - [A+R)' = A' B' ; {A+B+C)' = A'B'C ; and 
so on. 

43.05 Rule 5. -A+B ={(A+B)'}' = {A'B')' 

=AB , +(A , +A)B=AB'+B 
=A'B+A(B'+B = A'B+A. 

Similarly we get equivalents (with mutually inconsistent 

terms) for A+B+C,A+B+C+D, etc. 

43.06 Rule 11. - If A:B, then B':A'. Thus the implica- 
tions A:B and B':A' are equivalent, each following as a 
necessary consequence of the other. This is the logical prin- 
ciple of 'contraposition'. 

43.07 Rule 12. - If A:B, then AC:BC, whatever the state- 
ment C may be. 

43.08 Rule 13. - If A:*, B:$, C:y, then ABC: a^y, and so 
on for any number of implications. 

43.09 Rule 14. - If AB=0, then A:B' and B:A' . 

43.10 Definition 13. - The symbol A+B asserts that A does 
not imply B; it is thus equivalent to the less convenient 
symbol (A: By. 

43.11 Rule 15. - If A implies B and B implies C, then A 
implies C. 

43A2 Bute 16. - If A does not imply B 7 then B' does not 
imply A'; in other words, the non-implications A+B and 
B'+A f are equivalent. 

43.13 Rule 17. - If A implies R but does not imply C, then 
B does not imply C in other words, from the two premises 
A:B and A+C, we get the conclusion B+C. 

43.14 The following formulae are all either self-evident 
or easily verified, and some of them will be found useful in 
abbreviating the operations of the calculus : - 

(1) i< = ,0' = l; 

(2) l = l+a = l+a + b = l+a + b + c, etc. ; 

336 



PROPOSITIONAL LOGIC 



(3) {ab + a'b')' = a'b + ab' 

(a'b + ab')' = ab + a'b' 

(4 
(5 
(6 
(V 
(8 



(10 

(11 
(12 
(13 
(14 



a : a + b : a + b + c, etc. ; 

(a + A) (a+ D) (a + C) ... = a + A£C ...; 

(a : b) : a' + b; 

(a = b) = (a ib) {b : a); 

(a — b) : ab + a'b' ; 

(A :a) (5:6) (C : c) ... : (ABC ... : afo ...); 

(A : a) ( £ : b) ( C: c) ... : [A + £ + C ...:a + b + c + 

...); 

(A :aj) [B:x) {C : x) ... = (A + B + C + ... :x); 

(x:A) {x: B) (as C) ... = (x:i5C ...); 

(A :aj) + (B:«) + (C : x) + ... : {ABC ... :a?); 

(x : A) + [x : B) + (x : C) + ... : (x : A + B + C + ...). 

B. FREGE'S RULES OF INFERENCE 



One of Frege's most important intuitions concerned the distinc- 
tion between theorems and rules of inference. This is already to be 
found in the Begriffsschrift (43.15), where he uses only a single 
rule; in the Grundgesetze he adopts several for reasons of practical 
convenience, of which we give four: 

43.16 // the lower member of a proposition differs from a 
second proposition only in lacking the judgment-stroke, one 
can conclude to another proposition which results from the first 
by suppression of that lower member. 

'Lower member', i.e. antecedent. The sense is therefore: Given 
' | — if B, then A' and further ' | — B\ then we may suppress B in 
the conditional proposition to obtain ' | — A\ This is the modus 
ponendo ponens (22.04). 

43.17 A lower member may be exchanged with its upper 
member, if at the same time the truth-value of each is changed. 

Thus, given 'if B, then A\ one may write 'if not A, then not B' ; 
this is the rule of simple contraposition (31.17, cf. 43.22 [28]). 

43.18 // the same combination of symbols occurs as upper 
member in one proposition and lower member in another, one 
can conclude to a proposition in which the upper member of 
the second appears as upper member, and all lower members of 
the two, save the one mentioned, as lower members. Bui lower 
members which occur in both, need only be written once. 

337 



MATHEMATICAL VARIETY OF LOGIC 

Given 'if C, then B' and 'if B, then A\ one may write 'if C, then 
A\ This is the rule corresponding to the law of syllogism (cf. 31.18). 

43.19 // two propositions correspond in their upper members, 
while a lower member of one differs from a lower member of the 
other only in respect of a preceding negation-stroke, then we 
can conclude to a proposition in which the corresponding upper- 
member appears as upper member, and all lower members of the 
two, with the exception of the two mentioned, as lower members. 
Lower members which occur in both are only to be written down 
once. 

Frege's concrete example (43.20) is this: Given 'if e, then if not d, 
then: if b, then a' and 'if e, then if d, then: if b, then a', one may 
write: 'if e, then: if b, then a'. 

Lukasiewicz, deriving from Frege, formulates the difference 
between thesis and rule, and states the most important rule, as 
follows : 

43.21 A logical thesis is a proposition in which besides 
logical constants there occur only propositional or name- 
variables and which is true for all values of the variables that 
occur in it. A rule of inference is a direction which empowers 
the maker of inference to derive new theses on the basis of 
already admitted theses. Thus e.g. the laws of identity given 
above are logical theses, while the following 'rule of detach- 
ment' is a rule of inference : 

Whose admits as true the implication 'if a, then p' and the 
antecedent 'a' of this implication, has the right to admit as 
true also the consequent '[}' of this implication. 

Thus for Lukasiewicz, 'logical thesis' covers both axioms and 
derived propositions. 

C. PROPOSITIONAL LAWS FROM THE B EGR I FFSSCH R IFT 

Space prevents us from giving Frege's propositional schemata 
(corresponding to Lukasiewicz's theses) in the original symbolism; 
Instead, we translate some of them into Peano-Russellian. 

43.22 01. ud .bD a 

02. cD.bDaiDicDb.D.cDa 

03. bDa.Di.cD.bDaiDicDb.D.cDa 

04. b d a . d : c d . b d a : . d : . b d a : d : c d b . d . c 
d a 

05. bDaiDicDb.D.cDa 

338 




(x: 



'"(i-;::r"Mii-:) 



T '"ii:::r"'(v:i) 






a re 
a ~r 

a ^w 
a - c~/> 






An example of Frege's Symbolism taken from "Begriffsschrift", p. 56. 



PROPOSITIONAL LOGIC 



06. c d . b d a : . d : . c : d : d d b . d . d d a 

07. b d a : . d : . d d . c d b : d : d . d . c d a 

08. d d . b d a : d : b . d . d d a 

09. c d b : d : b d a . d . c d a 

10. e d . d d b : d a : . d . d d . e d b : d a 

11. CDb.D.aiD.bDCl 

12. d 3 : c . d . 6 d a : . d : . d : d : b . 

13. d d : c . d . b d a : . d : . b : d : d . 

14. n:.rf:D:c.D.ba::D::e 
d . d . c d a 

15. m.diDic.DiDo:::::!*:.:: 
d . c d a 



d . c d a 
d . c d a 
: . D : . 6 : d 



::>:</ 



16. 


e d : . d d : c d .b d a: : d : : e d : . d d : b d 


17. 


dD:c.D.bDa:.D:.cD:b.D.dDa 


18. 


cD.bDaiDi.dDciDibD.dDa 


19. 


dD.CDblDl.bDdlDldD.CDa 


20. 


e d : d d . c d b : : d : : b d a: . d : .e d : d . : 


21. 


dDb.DdlDl.dDC.DlCDb.DO. 


22. 


f d : : eD : . d d : c d .b d a:: . d :: . f :> : : < 




d : b d . c d a 


23. 


d d : c d . b d a : . d : : e d d . d : . c d : b . z 


24. 


CDd.DlCD.bDCl 


25. 


dD.CDdlDl.dDlCD.bDa 


26. 


b d . a d a 


27. 


a d a 


28. 


bDa.D.~aD~b 


29. 


cD.bDa:D:cD.~aD~b 


30. 


bD.cDa\D.CD.~aD~b 


31. 


~ ~ a d a 


32. 


~bDa,D.~aD~~b:Di~bDa.: 




Db 


33. 


~bDa.D.~aDb 


34. 


cD.~bDa:D:cD.~aDb 


35. 


cD.~bDa:D:~aD.cDb 


36. 


a d . <— ' a d b 


37. 


~CDb.Da:D.cDa 


38. 


~ a . d . a d b 


39. 


~aDa.D.~aDb 


40. 


~bD:~aDa.Da 


41. 


a d ~ ~ a 


42. 


~ ~ (a d a) 


43. 


~ a d a . d . a 



c d a 



c d a 



e d : . d 



e d a 



339 



MATHEMATICAL VARIETY OF LOGIC 

44. ~aDc.D:cDa.Da 

45. ~cDa.D.~aDc\D'. .~cDa:D:cDa.Da 

46. ~cDa.D:cDa.D.a 

47. ~cDb.D:.bDa.D:cDa.Da 

48. d d . ~ c d b : d : . b d a . d : c d a . d . d d a 

49. ~cDb.D\.cDa.D:bDa.Da 

50. CDa.D:.ha.D:~CDfi.Dfl 

51. d d . c d a: d ::b d a . d : . d d : ~ c d b . d a 

52. C ^d.D.f{c)Df{d) 

53. / (c) d : c = d . d f (d) 

54. c = c 

55. c = d . d . d = c 

56. d=c.D.f(d)Df{c) 

57. c =d.D .f{d)Df{c) 



D. WHITEHEAD AND RUSSELL 

Passing over Peano, we come now to the Principia Malhematica 
of Whitehead and Russell (Vol. 1, 1910). 

1. Primitive symbols and definition 

Besides variables, Frege's sign of assertion V and Peano's dots 
and brackets, the Principia uses only two undefined primitive 
symbols: '~' and 'v'. 'p' is read as 'not p', 'p v q' as 'p or q' the 
alternation being non-exclusive (43.23). 

Implication is defined : 

43.24 *1.01. p d q . = . ~ p v q Di. 

2. Axioms (Primitive Propositions) 

43.25 *1.1. Anything implied by a true elementary propo- 
sition is true. Pp. (Footnote : The letters "Pp" stand for "primi- 
tive proposition", as with Peano.) 

The above principle ... is not the same as "if p is true, then if 
p implies q, q is true". This is a true proposition, but it holds 
equally when p is not true and when p does not imply q. 
It does not, like the principle we are concerned with, enable 
us to assert q simply, without any hypothesis. We cannot 
express the principle symbolically, partly because any 
symbolism in which p is variable only gives the hypothesis that 
p is true, not the fact that it is true. 

43.26 *1.2. h :pv p. Dp Pp. 

This proposition states : "If either p is true or p is true, then 

340 



PROPOSITIONAL LOGIC 

p is true". It is called the "principle of tautology", and will be 
quoted by the abbreviated title of "Taut". It is convenient 
for purposes of reference, to give names to a few of the more 
important propositions; in general, propositions will be refer- 
red to by their numbers. 

43.27 *1.3. h :q. d . p v q Pp. 

This principle states: "If q is true, then 'p or 7' is true". 
Thus e.g. if q is "to-day is Wednesday" and p is "to-day is 
Tuesday", the principle states: "If to-day is Wednesday, then 
to-day is either Tuesday or Wednesday". It is called the 
"principle of addition". . . . 

43.28 *1.4. [- : p v q . d . q v p Pp. 

This principle states that "p or q" implies ll q or p". It states 
the permutative law for logical addition of propositions, and 
will be called the "principle of permutation". . . . 

43.29 *1.5. npv(gvr).3.?v(pvr)Pp. 

This principle states: "If either p is true, or 'q or r' is true, 
then either q is true, or 'p or r' is true". It is a form of the 
associative law for logical addition, and will be called the 
'associative principle'. . . . 

43.30 *1.6. \- :: . q d r . d : p v q . d . p v r Pp. 

This principle states: "If q implies r, then 'p or q' implies 
*p or p' ". In other words, in an implication, an alternative 
may be added to both premiss and conclusion without 
impairing the truth of the implication. The principle will be 
called the "principle of summation", and will be referred to 
as "Sum". 

3. Statement of proofs 

Two examples will explain the method of proof used in the 
Principia: 

43.31 *2.02. 1- :q.D . p Dq 

Dem. 

< — ' p 
Add \- : q . d . ~ pv q (1) 

P 
(1) .(*1.01) h :q. D.p Dq 

This is to be read: take 'Add', i.e. 43.27: 

q. D.p v q 

and in it substitute '~ p' for 'p'; we obtain 

341 



MATHEMATICAL VARIETY OF LOGIC 

As according to 43.24 '~ p v q' and 'p d q' have the same meaning, 
the latter can replace the former in ( 1 ) , which gives the proposition to 
be proved; it corresponds to the Scholastic verum sequitur ad quod- 
libel 

That proof is a very simple one; a slightly more complicated 
example is: 

43.32 *2.3. h :p v (q v r) . d . p v (r v q) 
Dem. 



Perm 
Sum 



q v r, r v q 



h : q v r • d . r v q: 

d t- : p v (q v r) . d .p v (r v q) 



4. Laws 



in 



the 



43.33 The most important propositions proved 
present number are the following : . . . 
*2.03. \-:pD~q.D.qD~p 
*2.15. \-:~pDq.D.~qDp 
*2.16. \-:pDq.D.~qD~p 
*2.17. h:~p^p.:.p:^ 

These four analogous propositions consitute the "prin- 
ciple of transposition" . . . 
*2.04. h :. p . d . q d r : d : q . d . p d r 
*2.05. h i.qDr.DipDq.D.pDr 
*2.06. h i.pDq.DiqDr.D.pDr 

These two propositions are the source of the syllogism in 
Barbara (as will be shown later) and are therefore called "the 
principle of the syllogism" . . . 
*2.08. \- . pD p 

I.e. any proposition implies itself. This is called the "prin- 
ciple of identity" . . . 
*2.21. \- : ~ p . d . p d q 

I.e. a false proposition implies any proposition. 

Next the Principia gives a series of laws concerning the logical 
product (43.34). At their head stand the two definitions: 

43.35 *3.01. p . q . = . — [~ p v — q) Df 
where "p . q" is the logical product of p and q. 
*3.02. pDqDr. = .pDq.qDrDf. 

This definition serves merely to abbreviate proofs. 

43.36 The principal propositions of the present number are 
the following: 



342 



PROPOSITIONAL LOGIC 

*3.2. h :. p , d : q . d . p . q 

I. e. "p implies that q implies p . ry", i.e. if each of two pro- 
positions is true, so is their logical product. 
*3.26. \- : p .q . d . p 
*3.27. h : p . q . p . q 

I.e. if the logical product of two propositions is true, then 
each of the two propositions severally is true. 
*3.3. h :. p . q . d . r : d : p . d . q d r 

I.e. if p and q jointly imply r, then p implies that q implies 
r. This principle (following Peano) will be called "exportation", 
because q is "exported" from the hypothesis . . . 
*3.31. h :. p . d . q d r : d : p . q . d . r . . . 
*3.35. I- : p . p d q . d . q 

I.e. "if p is true, and q follows from it, then q is true". This 
will be called the "principle of assertion". . . . 
*3.43. h i.pDq.por.Dip.D.q.r 

I.e. if a proposition implies each of two propositions, then 
it implies their logical product. This is called by Peano the 
"principle of composition". . . . 
*3.45. \- :. p d q . d : p . r . d . q . r 

I.e. both sides of an implication may be multiplied by a 
common factor. This is called by Peano the "principle of the 
factor" . . . 
*3.47. h i.pDr.qDs.Dip.q.D.r.s 

43.37 This proposition (*3.47), or rather its analogue for 
classes, was proved by Leibniz, and evidently pleased him, 
since he calls it "praeclarum theorema". 

43.38 *3.24. \- . ~ (p . ~ p) 

The above is the law of contradiction. 

Next equivalence is introduced (43.39) : 

43.40 When each of two propositions implies the other, we 
say that the two are equivalent, which we write "p = g". We 
put *4.01. p=q. = .pDq.qDpDi 

. . . two propositions are equivalent when they have the same 
truth-value. 

43.41 The principal propositions of this number are the 
following: 

M.l \-:pDq.=.~qo~p 

*4.11. \-: p=q.=.~p = ~q ... 

M.2. \- .p = p 

*4.21. y:p=q.=.q=p 

343 



MATHEMATICAL VARIETY OF LOGIC 

*4.22. \-:p=q.q=r.D.p=r 

These propositions assert that equivalence is reflexive, 
symmetrical and transitive. 
*4.24. h : p . = . p . p 
*4.25. h : p . = . p v p ... 

*4.3. h . p • 9 • 5 • 9 • P ••• 

*4.31. h:p v g. =.gvp ... 
M.32. h : (p . g) . r . = . p . (q . r) 
*4.33. \- : (p v q) v r . = . p v (q v r) ... 
*4.4. h:.p.gvr.=:p.g.v.p.r 
*4.41. h :. p . v . q , r : =.pvg.pvr 

The second of these forms (*4.41) has no analogue in 
ordinary algebra. 
*4.71. h :. p d g . = : p . = . p . 9 

i.e. p implies 9 when, and only when, p is equivalent to 
p . g. This proposition is used constantly; it enables us to 
replace any implication by an equivalence. 
*4.73. h :.q . d : p . = . p .q 

I.e. a true factor may be dropped from or added to a 
proposition without altering the truth-value of the proposition. 

43.42 *5.1. \- :p .q . d .p = q 

I.e. two propositions are equivalent if they are both true. . . . 
*5.32. i-:.p.D.g=r:=:p.^.=.p.r ... 
*5.6. 1- :. p . ~ q . d . r : = : p . d . q v r 

E.SHEFFER'S FUNCTOR 

In 1921 H. M. Sheffer showed that all propositional functors 
could be defined in terms of a single one, namely the 'stroke' ('|'). 
'p I g' means the same as 'not p or not q\ This was adopted in the 
second edition of the Principia (43.43). But there is another functor 
which will serve the same purpose and this was found by Peirce in 
1880. Here is his text: 

43.44 For example, x ju y signifies that x is f and y is f. 
Then (x ju y) ju z, or x ju y ju z, will signify that z is f , but that 
the statement that x and y are both f is itself f , that is, is false. 
Hence, the value of x ju x is the same as that of x\ and the 
value of x ju x ju x is f, because it is necessarily false; while 
the value of#Ju?/Ju;rju?/is only f in case x ju y is v; and 
(x ju x ju x) ju ( xju x ju x) is necessarily true, so that its value 
is v. 

With these two signs, the vinculum (with its equivalents, 

344 



PROPOSITIONAL LOGIC 



parentheses, brackets, braces, etc.) and the sign ju, which I 
will call the ampheck (from afjupyjxYJs , cutting both ways), all 
assertions as to the values of quantities can be expressed. 
Thus, 

Jb lo vL \J^f *L> \Xj «/• »X> tX 

Ob lo Jb \Xs Jb 

x : v : x is (x ju x ju x) ju (x ju x ju a:] 

Ob • Ob lo tA^ vA-/ e>L/ vA-» t>C 



- (xxyy) is {# ju y ju (a? ju y ju x ju y)} ju {(a; ju y x x ju y) 
ju a; ju ?/} 
a%cyi/ is xJuyju(xjuyjuXJuy) 
x = y is (x ju ?/ ju y) ju (x ju a; ju y) 
(x =y) is # ju y ju (a: ju a; ju y ju g) 
sv(/ is x Ju y Ju x Ju y 

x v y is (a; ju a; ju f/ ju y) ju (a; ju a? ju y ju y) [orxXy] 
a! v ?/ is (x Ju y Ju y) Ju (y Ju x Ju y) 

or [(y Ju x Ju x) ju (y Ju x ju x)] 
x . y is x Ju x ju y Ju y 
x . y is x Ju x ju y or x Ju y Ju y 



F.LUKASIEWICZ'S STATEMENT OF PROOFS 

Finally we give an example of the statement of pro-positional 
proofs in the very exact form developed by Lukasiewicz in 1920. 
The text quoted here dates from 1934, and is chosen for its brevity 
and clarity. 

43.45 The logical systems based on axioms are strictly 
formalized, i.e. the correctness of derivations can be checked 
without reference to or even knowledge of the meaning of 
the symbols used in them, provided only one knows the rules 
of inference. 

In illustration two examples of formalized proofs are given. 

a) Proof of the law of identity l Cpp' from the propositional 
axioms: 

1 CCpqCCqrCpr 

2 CCNppp 

3 CpCNpq 

1 q/Csq x 4 [substitution of l Csq' for 'q'] 

4 CCpCsqCCCsqrCpr 

4 s/Np x 5 [substitution of l Np' for V] 

5 CCpCNpqCCCNpqrCpr 



345 



MATHEMATICAL VARIETY OF LOGIC 

5 x C3 - 6 [detachment of 6 on the basis of 5 and 3] 

6 CCCNpqrCpr 

6 q/p, rjp x 7 [substitution of 'p' for 'q' and V] 

7 CCCNpppCpp 

7 x C 2 - 8 [detachment of 8 on the basis of 7 and 2] 

8 Cpp 



346 



IV. LOGIC OF TERMS 



44. PREDICATE LOGIC 



The matter so far discussed corresponds to the Megarian-Stoic 
doctrine and the Scholastic theory of consequeniiae, with the 
exception of the first interpretation of the Boolean calculus which 
is comparable to the Aristotelian (assertoric) syllogistic as being also 
a logic of terms. But in the second period of mathematical logic, i.e. 
mainly since Frege, two other forms of term-logic were developed. 
These are linked together in contrast to Boole's calculus (in its 
classical interpretation) essentially in the following respects: 

1. The Boolean calculus is purely extensional: it treats of classes, 
i.e. extensions of concepts. The calculus has no means at its disposal 
of dealing with meanings, much less of distinguishing them from 
classes. Now, on the other hand, in the new logic of terms, meaning 
and extension are treated in sharp distinction, so that we have two 
different doctrines: the logic of predicates, treating of meaning, and 
the logic of classes, treating of extensions. 

2. Within the logic of classes, the Boolean calculus has no place 
for individuals; in this respect it resembles the Aristotelian logic, or 
more exactly, its authors confused the relation of an individual to a 
class with that of one class to another, like Ockham and the 'classi- 
cal' logicians (confused, therefore, the Aristotelian relations of 
species and genus). These are now kept clearly distinct. 

3. The Boolean calculus expresses the Aristotelian quantifiers 
'all' and 'some' as operations on classes; it can therefore say, for 
instance, that all A is B, or that^l and B intersect. But it does this 
by means of relations between classes and the universe, without 
using the concept of the individual. Now, on the other hand, we 
meet one of the most interesting contributions made by mathemati- 
cal logic, viz. quantifiers 'all' and 'some' applied to individuals. 
In contrast to the Aristotelian tradition, these quantifiers are 
conceived as separate from the quantified function and its copula, 
and are so symbolized. 

We shall speak first of the development of the doctrine of quanti- 
fiers, which begins, so far as we know r , with Frege (1879). But the 
doctrine in the form given it by Frege, which is superior to all 
subsequent ones, remained at first quite unknown, until eventually 
Russell helped to establish it. Hence it must have been developed 
independently of Frege by Mitchell (1883), Peirce (1885) (cf. 44.021 
and Peano (1889). In the second place we give two texts about the 
concepts of free and bound variables, first explicitly introduced 
by Peano (1897), as it seems. Formal implication, which is closely 
connected with that, also derives from Peano (1889), but was 

347 



MATHEMATICAL VARIETY OF LOGIC 

thoroughly expounded by Peirce (1896). After that we make a 
selection of laws of predicate-logic from the Principia, and finally 
illustrate the theory of identity as found in Frege (1879) and Peirce 
(1885). 

A. QUANTIFIERS 
For the reasons stated we begin with Mitchell. 

1. Mitchell 
Mitchell writes : 

44.01 Let F be any logical polynomial involving class 
terms and their negatives, that is, any sum of products 
(aggregants) of such terms. Then the following are respecti- 
vely the forms of the universal and the particular proposi- 
tions: - 

All U is F, here denoted by F x 

Some U is F, here denoted by Fu. 
These two forms are so related that 

F 1 + Fu = oo 

F x Fu = ; 

that is, F 1 and Fu are negatives of each other; that is (F x ) = 
Fu. The two propositions F 1 and Fu satisfy the one equation 

F 1 F 1 = 0, 
and are 'contraries' of each other. Whence, by taking the 
negative of both sides, we get Fu + Fu= oo; that is Fu and Fu 
are 'sub-contraries' of each other. The line over the F in the 
above does not indicate the negative of the proposition, only 
the negative of the predicate, F. The negative of the propo- 
sition Fj is not F 1? but (FJ, which, according to the above = 
Fu. 

One might at first think that 'F 2 ' simply signifies 'all 1 are F\ but 
this is not so ; the subscript is not a subject, but precisely an index or 
quantifier which says that what 'F' represents holds universally, 
and correspondingly for l Fu\ This is the first step in the separation 
of the quantifier from the function, though this is still not too clear. 
In the continuation of that text, Mitchell introduces the symbols 
,IT and '£', but not as quantifiers. 

2. Peirce 

The doctrine is very clear in Peirce: 

44.02 We now come to the distinction of some and all, a 

348 



LOGIC OF TERMS 

distinction which is precisely on a par with that between 
truth and falsehood; that is, it is descriptive. 

All attempts to introduce this distinction into the Boolean 
algebra were more or less complete failures until Mr. Mitchell 
showed how it was to be effected. His method really consists 
in making the whole expression of the proposition consist 
of two parts, a pure Boolean expression referring to an indi- 
vidual and a Quantifying part saying what individual this 
is. Thus, if k means 'he is a king', and /i, 'he is happy', 

the Boolean (k + h) 

means that the individual spoken of is either not a king or 

is happy. Now, applying the quantification, we may write 

Any (k + h) 
to mean that this is true of any individual in the (limited) 
universe, or __ 

Some (k + h) 
to mean that an individual exists who is either not a king or 
is happy. 

44.03 In order to render the notation as iconical as possible 
we may use S for some, suggesting a sum, and II for all, 
suggesting a product. Thus HiiXi means that x is true of some 
one of the individuals denoted by i or 

TtiXi = Xi + X) + Xk + etc. 

In the same way, U 2 x 2 means that x is true of all these indi- 
viduals, or 

UiXi = XiXjXk, etc. 

If z is a simple relation, UiUjXij means that every i is in this 
relation to every /, SiII/£Cy means that some one i is in this 
relation to every /, UjHixtj that to every / some i or other is 
in this relation, £iE/xc;; that some i is in this relation to 
some /. It is to be remarked that Htxt and Utxi are only 
similar to a sum and a product; they are not strictly of that 
nature, because the individuals of the universe may be 
innumerable. 

We have already met the basic idea of this text, having seen it 
explicitly formulated by Albert of Saxony (34.07), so that this is a 
re-discovery. Quite new, on the other hand, is the clear separation 
of the quantifier from the formula quantified. 

3. Peano 

Peirce's notation was adopted by Schroder (44.04) and today is 
still used in Lukasiewicz's symbolism. But Peano's is more widely 

349 



MATHEMATICAL VARIETY OF LOGIC 

established, since its essentials were taken over in the Principia. 
The Italian logician introduces it in these words: 

44.05 If the propositions a, b, contain undetermined 
beings, such as x, y, ..., i.e. if there are relationships between 
the beings themselves, then a d x, y , ... b signifies: whatever 
x, y, ..., may be, b is deduced from the proposition a. To 
avoid risk of ambiguity, we write only d instead of d x,y, .... 

4. Frege 

Now we come to Frege's Begriffsschrift: 

44.06 In the expression for a judgment, the complex symbol 
to the right of | — may always be regarded as a function of 
one of the symbols that occur in it. Lei us replace the argument 
with a Gothic letter, and insert a concavity in the content-stroke, 
and make this same Gothic letter stand over the concavity: e.g.: 

h-\J>— O (a) 
This signifies the judgment that the function is a fact whatever 
we take its argument to be. A letter used as a functional symbol, 
like O in (A), may itself be regarded as the argument of a 
function; accordingly, it may be replaced by a Gothic letter, 
used in the sense I have just specified. The only restrictions 
imposed on the meaning of a Gothic letter are the obvious 
ones: (i) that the complex of symbols following a content- 
stroke must still remain a possible content of judgment 
(41.11); (ii) that if the Gothic letter occurs as a functional 
symbol, account must be taken of this circumstance. All 
further conditions imposed upon the allowable substitutions for 
a Gothic teller must be made part of the judgment. From such a 
judgment, therefore, we can always deduce any number we 
like of judgments with less general content, by substituting 
something different each time for the Gothic letter; when 
this is done, the concavity in the content-stroke vanishes 
again. The horizontal stroke that occurs to the left of the 
concavity in 

is the content-stroke for (the proposition) that $ (a) holds 
good whatever is substituted for a; the stroke occurring to the 
right of the concavity is the content-stroke of O (a) - we 
must here imagine something definite substituted for a. 

By what was said before about the meaning of the judg- 
ment-stroke, it is easy to see what an expression like 

350 



LOGIC OF TERMS 

—&— * (a) 

means. This expression may occur as part of a judgment, 
as in 

H-O— X(a), l—l A 

It is obvious that from these judgments we cannot infer 
less general judgments by substituting something definite 
for a, as we can from 

I— vir- X[a) 
h~\^ — X (a) serves to deny that X (a) is always a fact 
whatever we substitute for a. But this does not in any way 
deny the possibility of giving a some meaning A such that 
X (A) is a fact. I — | A 

means that the case in which — \^y — J^T (a) is affirmed and 
A denied does not occur. But this does not in any way deny 
the occurrence of the case in which X (A) is affirmed and A 
denied; for, as we have just seen, X (A) may be affirmed and, 
nevertheless, — \^j — X(a) denied. Thus here likewise we 
cannot make an arbitrary substitution for a without prejudice 
to the truth of the judgment. This explains why we need the 
concavity with the Gothic letter written on it; it delimits 
the scope of the generality signified by the letter. A Gothic letter 
retains a fixed meaning only within its scope; the same Gothic 
letter may occur within various scopes in the same judgment, 
and the meaning we may ascribe to it in one scope does not 
extend to any other scope. The scope of one Gothic letter may 
include that of another, as is shown in 

B(a, e) 

In this case different letters must be chosen; we could not 
replace e by a. It is naturally legitimate to replace a Gothic 
letter everywhere in its scope by some other definite letter, 
provided that there are still different letters standing where 
different letters stood before. This has no effect on the con- 
text. Other substitutions are permissible only if the concavity 
directly follows the judgment-stroke, so that the scope of the 
Gothic letter is constituted by the content of the whole 
judgment. Since this is a specially important case. I shall 
introduce the following abbreviation: an italic letter is always 



351 



MATHEMATICAL VARIETY OF LOGIC 

to have as its scope the content of the whole judgment, and this 
scope is not marked out by a concavity in the content- 
stroke. If an italic letter occurs in an expression not preceded 
by a judgment-stroke, the expression is senseless. An italic 
letter may always be replaced by a Gothic letter that does not yet 
occur in the judgment; in this case the concavity must be 
inserted immediately after the judgment-stroke. E.g. for 

t—X{a) 
we may put 

\-y^r-X(a) 
since a occurs only in the argument-position within X (a). 
Likewise it is obvious that from 

h— <—<& (a) 




we may deduce 

if A is an expression in which a does not occur, and a occupies 
only argument-positions in O (a). If — \^y — O (a) is denied, 
we must be able to specify a meaning for a such that (a) 
is denied. Thus if O (a) were denied and A affirmed, 

we should have to be able to specify a meaning for a such 
that A was affirmed and O (a) denied. But since we have 

hy-O(a) 
] — A 
we cannot do so; for this formula means that whatever a 
may be the case in which O (a) would be denied and A affirmed 
does not occur. Hence we likewise cannot both deny — y^ — 
O (a) and affirm A : i.e. 

hvy- 0(a) 

1 A. 

44.07 We may now consider certain combinations of 
symbols. 

1-rvV— X(a) 
means that we can find something, say A, such that X (A) 
is denied. We may thus render it as: 'there are some things 
that have not the property X\ 
The sense of 

H-Ot- X(<x) 
is different. This means: 'Whatever a may be, X (a) must 
always be denied', or 'there is not something with the property 
X\ or (calling something that has the property X, a X) 
'there is no X\ 

352 



LOGIC OF TERMS 

A (a) is denied by 
A (a) 
This may therefore be rendered as 'there are A V. 

This an extremely important text. Frege quite clearly teaches 
the separation of quantifier and quantified function, introduces the 
concept of 'bound variable' (though without so naming it), defines 
the existential quantifier, and investigates what happens when there 
is more than one quantifier. 



B. APPARENT VARIABLES 

1. Peano 

The expressions 'real' and 'apparent variable' derive from Peano 

(1897): 

44.08 In these explanations we say that a letter in a 
formula is real or apparent, as the case may be, according as 
the value of the formula depends or does not depend on the 

name of this letter. Thus in / x m dx the letter a; is apparent and 
the letter m real. All letters occuring in a theorem are apparent, 
since its truth is independent of the names of the letters. 

The expressions 'free' and 'bound variable' are used today in just 
the same sence. 

2. Whitehead and Russell 

It is strange that the problem of quantification is only super- 
ficially touched on in Russell's Principles of Mathematics (1903). But 
it is thoroughly treated in Principia Mathematica (1910), from the 
introduction to which comes the following passage: 

44.09 Corresponding to any propositional function <px, 
there is a range, or collection, of values, consisting of all the 
propositions (true or false) which can be obtained by giving 
every possible determination to x in <px. A value of x for which 
cpx is true will be said to "satisfy" <px. Now in respect to the 
truth or falsehood of propositions of this range three important 
cases must be noted and symbolized. These cases are given 
by three propositions of which one at least must be true. 
Either (1) all propositions of the range are true, or (2) some 
propositions of the range are true, or (3) no proposition of the 
range is true. The statement (1) is symbolized by "(x) . cpz", 
and (2) is symbolized by (3a;) . <px. . . . The symbol "(a;) . yx" 
may be read "cpx always", or "cp,r is always true", or "qxc is 

353 



MATHEMATICAL VARIETY OF LOGIC 

true for all possible values of x". The symbol u (3x) . cpx" may 
be read ''there exists an x for which cpx is true", or "there exists 
an x satisfying cpa?", and thus conforms to the natural form 
of the expression of thought. 

44.10 Apparent variables. The symbol "(x) . <pa:)" denotes one 
definite proposition, and there is no distinction in meaning 
between "(x) . cp#" and "(*/) . <p*/" when they occur in the same 
context. Thus the "x" in "(x) . cp#" is not an ambiguous consti- 
tuent of any expression in which "(x) . (px" occurs; ... The 
symbol "(x) . 92:" has some analogy to the symbol 

"fl ?(*) dx" 

... The x which occurs in "(x) . yx" or ("3a;) . yx" is called 
(following Peano) an 'apparent variable' ... A proposition in 
which x occurs as an apparent variable is not a function of 
x. Thus e.g. "(x) . x = x" will mean "everything is equal to 
itself". This is an absolute constant, not a function of a 
variable x. 

C. FORMAL IMPLICATION 

The theory of what Russell calls 'formal implication' is closely 
connected with that of quantification. It is already suggested by 
Peano (44.05) in the formula a d x b, but Peirce is the first to explain 
it clearly: 

44.11 Now let us express the categorical proposition, 
'Every man is wise'. Here, we let mi mean that the individual 
object i is a man, and wt mean that the individual object i 
is wise. Then, we assert that, 'taking any individual object 
of the universe, i, no matter what, either that object, i, is 
not a man or that object, 1, is wise' ; that is, whatever is a 
man is wise. That is, 'whatever i can indicate, either rrn is 
not true or wi is true! The conditional and categorical 
propositions are expressed in precisely the same form; and 
there is absolutely no difference, to my mind, between them. 
The form of relationship is the same. 

Russell writes: 

44.12 For the technical study of Symbolic Logic, it is 
convenient to take as a single indefinable the notion of a 
formal implication, i.e. of such propositions as "x is a man 
implies x is a mortal, for all values of #"- propositions whose 

354 



LOGIC OF TERMS 

general type is: "cp(oj) implies ty(x) for all values of z", where 
9 (x), ty (x), for all values of x, are propositions. The analysis 
of this notion of formal implication belongs to the principles 
of the subject, but is not required for its formal development. 

The suggestion for study which Russell makes here, has not been 
taken up so far as we know. 

44.13 It is to be observed that "x is a man implies a: is a 
mortal" is not a relation of two propositional functions, but 
is itself a single propositional function having the elegant 
property of being always true. For "x is a man" is, as it 
stands, not a proposition at all, and does not imply anything; ... 



D. LAWS OF ONE-PLACE PREDICATES 

After what has been said, we can here confine ourselves to some 
remarks and examples from the Principia: 

44.14 We have proved in *3.33 that 
poq.qDr.D.pDr. 

Put p = Socrates is a Greek, 

q = Socrates is a man, 

r = Socrates is a mortal. 
Then we have "if 'Socrates is a Greek' implies 'Socrates is 
a man', and 'Socrates is a man' implies 'Socrates is a 
mortal', it follows that 'Socrates is a Greek' implies 'Socra- 
tes is a mortal'". But this does not of itself prove that if all 
Greeks are men, and all men are mortals, then all Greeks 
are mortals. 
Putting 

cp# . = . x is a Greek, 

4* x . = . x is a man, 

X x . = . x is a mortal, 
we have to prove 

(x) . 9 x d 4 x : (x) . ^ x d i x : d : (x) . cp x d x x - • • • 
We shall assume in this number, . . . that the propositions 
of *l-*5 (cf. 43.26-42) can be applied to such propositions 
as (x) . cp# and (3x) . <px. . . . We need not take (3x) . (?x as a 
primitive idea, but may put 
*10.01. (3os) . 95c . = . ~ (x) . — <?x Df 

44.15 *10.1. h : (x) . <px . d . yy 

I.e. what is true in all cases is true in any one case. 

355 



MATHEMATICAL VARIETY OF LOGIC 

*10.11. If <py is true whatever possible argument y may be, 
then (a?) . 9 x is true. . . . 

* 10.23. I- :. (x) . 9 x d p . = : (3a?) . <pa? . D . p 

I.e. if cpx always implies p, then if <px is ever true, p is true. 

* 10.24. h : cpy . d . (3x) . 9a? 

/.e. if <py is true, then there is an x for which <px is true. This 
is the sole method of proving existence-theorems. 
*10.27. h :. (z) . <pz d tyz . d: (z) . (pz . d . (z) . 92 

/.e. if cpz always implies tyz, then "9Z always" implies "^z 
always". 

44.16 *10.26. h :. (z) . 9Z d <|;z : 9a? : d . <|/a? 

This is one form of the syllogism in Barbara. E.g. put 
(pz . = . z is a man, ^z . = . z is mortal, x = Socrates. Then the 
proposition becomes: "If all men are mortal, and Socrates is 
a man, then Socrates is mortal". 

Another form of the syllogism in Barbara is given in *10.3 
(cf. 44.17). The two forms, formerly wrongly identified, were 
first distinguished by Peano and Frege. . . . 
M0.271. I- :. (z) . 9 z = ^z . d : (z) . 9Z . = . (z) . <\>z 
*10.28. h :. (x) . <pa? d tyx . d : (3a?) . <px . d . (3a;) . tyx 
*10.281. h :. (x) . (px = tyx , d : (3a?) . 9a? . = . (3a?) . ^a? 

44.17 *10.3.h :. (a?) . 9a? d <J;a? : (a?) . tyx d jx : d . (a?) .cpa? dx # 

44.18 * 10.35. h :. (3a?) . p . 9a? . = : p : (3a?) . 92 

44.19 *10.42. h :. (3a?) . <px . v . (3a?) . +a? : . ( 3a?). cpx v <\>x 
*10.5.h :. (3a?) . 9a? . ^a? . <]ix . d : (3a?) . 9a? : (3a?) . tyx 

44.20 The converse of the above proposition is false . . . 
while * 10.42 states an equivalence. . . . 

44.21 * 10.51. h :. ~{ (3a?) . 9a? . ^a? }. = : 9a? . Dx . ~ tyx 

The distinction between the two forms of syllogism in Barbara 
consists, when expressed in Aristotelian terms, in the fact that in 
44.17 the minor premiss is a universal proposition, in 44.16 a singu- 
lar one. The false identification of the two, of which Russell speaks, 
is not Aristotelian, but first found in Ockham (34.01). 



E. LAWS OF MANY-PLACE PREDICATES 

The concept of a many-place function (42.07 ff.) led Frege to 
plural quantification. This is not to be confused with quantification 
of the predicate (36.15 ff.), since what is here quantified is not the 
predicate, but two parts of the subject of predication. 

We cite the most important laws which inter-relate such propo- 
sitions from the Principia. 

356 



LOGIC OF TERMS 



44.22 *11.1. h : (a:, y) . 9 (a:, y) . _ . (,y, *) . 9 (a, </) 

44.23 * 11.23. h : (3a:, y) . 9 (x, y) . m . (3y, x) . 9 (a:, y) 

44.24 * 11.26. I- :. (3a:) : (y) . 9 (*, If) s D : (</) : (3a:) . 9 (*,</) 
. . . Note that the converse of this proposition is false. 3. g. let 
9 (x, y) be the propositional function 'if y is a proper fraction, 
then x is a proper fraction greater than y\ Then for all values 
of y we have (3a:) . 9 (x, y), so that (y) : (3a:) . 9(3:, y) is 
satisfied. In fact l (y) : (3a:) . 9(3:, y)' expresses the proposition: 
'If y is a proper fraction, then there is always a proper fraction 
greater than y\ But '(3a:) : (y) . 9(3:, y)' expresses the propo- 
sition: 'There is a proper fraction which is greater than any 
proper fraction', which is false. 

44.24 is a re-discovery of a theorem from the doctrine of com- 
pounded and divided sense (29.10 ft". ) 



F. IDENTITY 

One logical two-place predicate which has special importance is 
identity. In the Boolean period it was introduced without defini- 
tion. In later mathematical logic it was defined — conformably with 
an idea of Aristotle's (16.13) — by means of one-place predicates 
and implication or equivalence. Leibniz formulated this thought in 
his principe des indiscernables, on ontological grounds and, as it 
seems, without the help of a mathematical-logical symbolism. The 
first definition in mathematical logic is to be found in Frege. 

44.25 Equality of content differs from conditionality and 
negation by relating to names, not to contents. Elsewhere, 
names are mere proxies for their content, and thus any phrase 
they occur in just expresses a relation between their various 
contents; but names at once appear in propria persona so 
soon as they are joined together by the symbol for equality of 
content; for this signifies the circumstance of two names' 
having the same content. Thus, along with the introduction 
of a symbol for equality of content, all symbols are necessarily 
given a double meaning - the same symbols stand now for 
their own content, now for themselves. At first sight this 
makes it appear as though it were here a matter of something 
pertaining only to expression, not to thought; as though we 
had no need of two symbols for the same content, and there- 
fore no need of a symbol for equality of content either. 
In order to show the unreality of this appearance, I choose the 
following example from geometry. Let a fixed point A lie 

357 



MATHEMATICAL VARIETY OF LOGIC 

on the circumference of a circle, and let a straight line rotate 
around this. When this straight line forms a diameter, let us 
call the opposite end to A the point B corresponding to this 
position. Then let us go on to call the point of intersection of 
the straight line and the circumference, the point B corres- 
ponding to the position of the straight line at any given time ; 
this point is given by the rule that to continuous changes in 
the position of the straight line there must always correspond 
continuous changes in the position of B. Thus the name B has 
an indeterminate meaning until the corresponding position 
of the straight line is given. We may now ask: What point 
corresponds to the position of the straight line in which it is 
perpendicular to the diameter? The answer will be: The point 
A. The name B thus has in this case the same content as the 
name A; and yet we could not antecedently use just one name, 
for only the answer to the question justified our doing so. 
The same point is determined in a double way: 

(1) It is directly given in experience; 

(2) It is given as the point B corresponding to the 
straight line's being perpendicular to the diameter. 

To each of these two ways of determining it there answers a 
separate name. The need of a symbol for equality of content 
thus rests on the following fact: The same content can be 
fully determined in different ways; and that, in a particular 
case, the same content actually is given by two ways of deter- 
mining it, is the content of a judgment. Before this judgment 
is made, we must supply, corresponding to the two ways of 
determination, two different names for the thing thus deter- 
mined. The judgment needs to be expressed by means of a 
symbol for equality of content, joining the two names together. 
It is clear from this that different names for the same content 
are not always just a trivial matter of formulation; if they go 
along with different ways of determining the content, they 
are relevant to the essential nature of the case. In these cir- 
cumstances the judgment as to equality of content is, in 
Kant's sense, synthetic. A more superficial reason for intro- 
ducing a symbol for equality of content is that sometimes it 
is convenient to introduce an abbreviation in place of a 
lengthy expression; we then have to express equality of 
content between the abbreviation and the original formula. 

(— (A . B) 
is to mean: the symbol A and the symbol B have the same 

358 



LOGIC OF CLASSES 

conceptual content, so that A can always be replaced by B and 
conversely. 

This analysis is remarkably like that of Thomas Aquinas (29.02). 
Only, as is evident, Frege conceives identity as a relation between 
two names, and so defines it in a meta-language. Later mathematical 
logicians have not followed him in this, but have thought of identity 
as a relation between objects. This modern definition (which also 
corresponds to Aristotle's thought) is to be found first in Peirce: 

44.26 We may adopt a special token of second intention, 
say 1, to express identity, and may write l/y. But this relation 
of identity has peculiar properties. The first is that if i and / 
are identical, whatever is true of i is true of /. This may be 
written 

IL IT; {lij + Xi + Xj } . . . 
The other property is that if everything which is true of 
i is true of /, then i and / are identical. This is most naturally 
written as follows: Let the token, q, signify the relation of a 
quality, character, fact, or predicate to its subject. Then the 
property we desire to express is 

IL Uj £ft [lij + qki qkj)' 
And identity is defined thus 

lij = lift [qki qkj + qki qkj). 
That is, to say that things are identical is to say that every 
predicate is true of both or false of both. It may seem circui- 
tous to introduce the idea of a quality to express identity; 
but that impression will be modified by reflecting that qki qkj* 
merely means that i and / are both within the class or collec- 
tion k. If we please we can dispense with the token q, by 
using the index of a token and by referring to this in the 
Quantifier just as subjacent indices are referred to. That is to 
say, we may write 

lij = IIx [xi Xj + Xi Xj). 



§45. THE LOGIC OF CLASSES 

The so-called 'pure' logic of classes, i.e. the theory of relations 
between classes, was developed as the first interpretation of the 
Boolean calculus. But as already stated, this had no means of 
expressing the relation of an individual to a class to which it belonged. 

* Reading qki qkj instead of qki qjk 

359 



MATHEMATICAL VARIETY OF LOGIC 

Further, the concept of class was taken as primitive, not defined. 
Here the later growth of mathematical logic brought two important 
novelties: first, the introduction of the concept of the relation 
between an individual and a class as distinct from that of class 
inclusion; second, the reduction of classes to properties (predicates), 
through definition. 

A. INDIVIDUAL AND CLASS. CONCEPT OF ELEMENT 

The first of these novelties is to be found, like so many others, 
originating in Frege's Begriffsschrift (1879). Just ten years later it 
appears also in Peano, who did not then know the Begriffsschrift. 
Here again Frege's work, though earlier and better than Peano's, 
remained without influence till Russell (1903). For this reason we 
begin with Peano: 

45.01 Concerning classes 

By the symbol K is signified a class, or aggregation of 
beings. 

The symbol e signifies is (est). Thus a z b is read a is a b; 
a z K signifies a is a class; a s P signifies a is a proposition. 

In place of - (a z b) we shall write a -e b; the symbol -s 
signifies is not; i.e. : 

a -s b . = : -a z b. 

The symbol a, b, c z m signifies: a, b, and c are m; i.e.: 

a, b, c z m . = :azm.bzm.czm. 

Let a be a class; then -a signifies the class constituted by 
the individuals which are not a. 

a z K . d : x z -a . = . x -z a. 

B. MEANING AND EXTENSION 

Only then could the problem of priority as between meaning and 
extension be posed with full accuracy. Frege was a convinced 
intensionalist, i.e. he emphatically maintained the priority of 
meaning - of concept, in his terminology - over extension, i.e. 
over class. His clearest formulation of this was in a communication 
to Jourdain in 1910 - later, then, than Russell's statement of the 
problem of antinomies: 

45.02 In my fashion of regarding concepts as functions, 
we can treat the principal parts of logic without speaking 
of classes, as I have done in my Begriffsschrift, and the 
difficulty does not come into consideration. Only with 
difficulty did I resolve to introduce classes (or extents of 
concepts), because the matter did not appear to me quite 

360 



LOGIC OF GLASSES 

secure - and rightly so, as it turned out. The laws of numbers 
are to be developed in a purely logical manner. But the 
numbers are objects, and in Logic we have only two objects, 
in the first place : the two truth values. Our first aim, then, was 
to obtain objects out of concepts, namely, extents of concepts 
or classes. By this I was constrained to overcome my resistance 
and to admit the passage from , concepts to their extents. 
And, after I had made this resolution, I made a more extended 
use of classes than was necessary, because by that many 
simplifications could be reached. I confess that, by acting 
thus, I fell into error of letting go too easily my initial doubts, 
in reliance on the fact that extents of concepts have for a 
long time been spoken of in Logic. The difficulties which are 
bound up with the use of classes vanish if we only deal with 
objects, concepts, and relations, and this is possible in the 
fundamental part of Logic. The class, namely, is something 
derived, whereas in the concept - as I understand the word - 
we have something primitive. Accordingly, also the laws of 
classes are less primitive than those of concepts, and it is 
not suitable to found Logic on the laws of classes. The primi- 
tive laws of Logic should contain nothing derived. We can, 
perhaps, regard Arithmetic as a further developed Logic. But, 
in that, we say that in comparison with the fundamental 
Logic, it is something derived. On this account I cannot think 
that the use of arithmetical signs ('+', '-', ':') is suitable in 
Logic. The sign of equality is an exception; in Arithmetic 
it denotes, at bottom, identity, and this relation is not 
peculiar to Arithmetic. It must be doubtful a priori that it is 
suitable to constrain Logic in forms which originally belong 
to another science. 

We saw that Peano introduced the concept of the relation between 
individual and class, i.e. the concept of element (s). On the most 
obvious interpretation he conceived the matter extensionally. 

In 1903 Russell formulated the problem thus: 

45.03 Class may be defined either extensionally or inten- 
sionally. That is to say, we may define the kind of object 
which is a class, or the kind of concept which denotes a class: 
this is the precise meaning of the opposition of extension 
and intension in this connection. But although the general 
notion can be defined in this two-fold manner, particular 
classes, except when they happen to be finite, can only be 

361 



MATHEMATICAL VARIETY OF LOGIC 

defined intensionally, i.e. as the objects denoted by such 
and such concepts. I believe this distinction to be purely 
psychological: logically, the extensional definition appears 
to be equally applicable to infinite classes, but practically, 
if we were to attempt it, Death would cut short our laudable 
endeavour before it had attained its goal. 

Points to be noted here, and often neglected, are: (1) no modern 
logician is extensionalist in the sense of adopting exclusively a logic 
of classes, without a logic of predicates. (2) For class-logic itself 
there are two bases possible, an extensional and an intensional. 
(3) Russell - but not all logicians mentioned, Frege for instance - is 
of the opinion that these bases are to be theoretically equated. (4) 
But he admits that in practice the foundation of class-logic, and so 
of the extensional aspect of term-logic, must be intensional. 

From the concept of extensionality touched on here, we must 
distinguish another, mentioned in the Principia in connection with 
propositional functions. For the Principia uses an intensional 
method of defining classes, in the sense that it uses the concept of 
propositional function in which there occurs a name of a property - 
and so a predicate. But a propositional function itself can be con- 
ceived either intensionally or extensionally. The most important 
text on this subject in the Principia is this: 

45.04 When two functions are formally equivalent, we shall 
say that they have the same extension. ... Propositions in which 
a function 9 occurs may depend, for their truth-value, upon 
the particular function 9, or they may depend only upon 
the extension of 9. In the former case, we will call the propo- 
sition concerned an intensional function of 9; in the latter 
case, an extensional function of 9. Thus, for example, (x) . 9 x 
or (3x) . yx is an extensional function of 9, because, if 9 
is formally equivalent to <|>, i.e. if 9 x . = x . tyx, we have (x) . 
cpx . = . (x) . <\>x and (3x) . <px . = . (3a;) . tyx. But on the 
other hand 'I believe (x) . tpa;' is an intensional function, 
because, even if yx . = x . tyx, it by no means follows that I 
believe (x) . tyx provided I believe (x) . <px. 

C. THE PLURAL ARTICLE 

Frege was the first to state a purely intensional definition (in the 
first sense) of classes. He could do this because he had at his disposal 
a symbol which transformed a function into its range of values: so 
that l F belongs to all x" becomes 'the x-s to which F belongs'. 
Frege's text is as follows: 

362 



LOGIC OF CLASSES 

45.05 Our symbolism must also be able to show the trans- 
formation of the universality of an equation into an equation 
between ranges of values. Thus, for instance, for 

' — \£j — a 2 - a= a. {a- 1)' 
I write 'k {z 2 - z) = a (a . (a - 1))' 

where by l k (z 2 - e)' I understand the range of values of the 
function I 2 - ?, by '<* (a . (a - 1))' that of \ . [I - 1). Similarly 
I (s 2 = 4) is the range of values of the function £ 2 = 4, or, as 
we can also say, the comprehension of the concept square 
root of four. 

After Frege, and evidently independently of him, Peano developed 
a similar idea. His theory is not entirely intensional, in that he 
uses the concept of element (z) in his definition of classes. 

45.06 Let a be a K(i.e. class). Let us write the symbol xz 
before the symbol a; by the convention P2 (cf. 45.01) we 
obtain the proposition 

x z a 
which contains the variable letter x. Now we make the con- 
vention that in writing the symbol x z before this proposition, 
the formula 

x z (x z a) 
again represents the class a. 

This convention is usefully applied which the proposition 
containing the variable letter x is not yet reduced to the 
form x z a. Let p be a proposition containing the variable 
letter x; the formula x z p indicates the class of '.r-es which 
satisfy the condition p\ 

The symbol x z can be ready by the phrase 'the x-es which'. 

Example : 1 z x z (x 2 - 3x + 2 = 0) 
'unity is a root of the equation in parentheses'. 

Let us note that in the formula x z p the letter x is apparent; 
the value of the formula does not change if we substitute for 
the letter x another letter y, in the symbol x z and in the pro- 
position p. 

D. DEFINITION OF CLASSES BY FUNCTIONS 

In the text just quoted Peano comes near to an intensional 
definition of classes, when he speaks of a condition p. This idea 
receives explicit formulation in the Principia, where Russell and 
Whitehead write 'a?' for 'the x-s which . . .'. Their basic definition is 
this: 

363 



MATHEMATICAL VYRIETY OF LOGIC 

45.07 *20.03. Cls == &{ (3 9) . a = z (9 ! z) } Df 

This is the definition of the class of classes, and so (on 
an intensional interpretation) of the concept of class as such. 
The class of classes, symbolized by 'Cls', is identical with 
those ol -s (a) for which there is a property cp such that a is 
identical with the z-s of which 9 is true. (The point of excla- 
mation means that the last function has the name of an 
individual as argument and is elementary.) 

The sense of this rather complicated and abstract definition 
becomes clear in this law which is deduced with its aid : 

45.08 *20.3. h : x e z (tyz) . = . <|a 

i.e. 'x is an element of the class of the z-s such that tyz, if and 
only if tyx'. 



E. PRODUCT AND INCLUSION OF CLASSES 

Peano succeeded in defining relations between classes by means 
of the definite article and quantifiers, propositional functions being 
presupposed. We cite two examples of such definitions from the 
Formulaire (1897) and a discussion of the difference between being 
an element and being included in a class. 

45.09 Let a and b be classes; by a n b we indicate the class 
xz (x z a . x z b). 

45.10 Thus the logical product of Ks (classes) has been 
defined by the logical product of P (proposition)s, the latter 
being taken as a primitive idea. 

45.11 Implication and Inclusion. 

Let a and b be Ks. In place of the proposition 
x z a . Dx . x z b 
'whatever x may be, if it is an a, it is also a b\ we shall write 
the formula, which no longer contains the apparent letter x, 

a d b 
which can be read: 'all a is b\ or 'the class a is contained in 
the class b\ 

45.12 The symbols s and d, which we have introduced, 
and the symbol =, well known to the reader . . . have dif- 
ferent significations, though they sometimes correspond to the 
same words in language. E.g.: 

'7 is the sum of 3 and 4' can be rendered as '7 = 3 + 4' . . . 
'7 is a prime number' can be rendered as '7 z Np' . . . 

364 



LOGIC OF CLASSES 



'All multiples of 6 are multiples of 3' can be rendered by 
'JV x 6 d N x 3'. 

These symbols also obey different laws. 



§ 46. EXISTENCE 

Within mathematical logic, at least since Schroder, there have 
been two sets of problems concerning existence. The first is in 
connection with the question of the so-called null class, already 
raised in the Middle Ages; for with the admission of such a class 
certain difficulties arise in the Aristotelian syllogistic. The second 
is concerned with propositions in which existence is ascribed to an 
individual, and has led to the important logical doctrine of descrip- 
tion as found in Frege and Russell. 

A. THE NULL CLASS 

We treat here of the essentials on both these points. 

The concept of the null class, containing no elements, was tacitly 
introduced in Boole's Analysis, in that Boole simply took over 
zero from algebra as the symbol of such a class. This occurs as 
follows, in his interpretation of the proposition 'all Xs are Ys\ 

46.01 As all the Xs which exist are found in the class Y, 
it is obvious that to select out of the Universe all Ys, and 
from these to select all Xs, is the same as to select at once 
from the Universe all Xs. 
Hence xy = x, 

or s(l-y) =0, (4). 

There is then a dissymmetry in Boole, since he introduces the 
universal class (1) explicitly (40.05), but, so to say, tacitly smuggles 
in the null class (0). This dissymmetry is removed in Schroder, both 
concepts being introduced side by side in parallel fashion. 

46.02 Two special domains are now to be introduced into 
the algebra of logic, for the names of which . . . the numerals 
and 1 recommend themselves. These too we shall explain 
by means of the relation-symbol of inclusion, for the 



Definition (2x) of the 'iden- 
tically null' 



Definition (2+) of the 'iden- 
tically one' 



follows from our positing as universally valid, i.e. as to be 

admitted for every domain a of our manifold, the subsumption 

0=0 | a£l 

365 



MATHEMATICAL VARIETY OF LOGIC 



That means to say 



is what we call a domain 
which stands in the relation 
of inclusion to every domain 
a, which is contained in every 
domain of the manifold. 



1 is what we call a domain 
to which every domain a 
stands in the relation of in- 
clusion, in which every do- 
main of the manifold is con- 
tained. 

46.03 Purely on didactical grounds . . . meanwhile, the 
interpretation which the symbols and 1 will have, may be 
already briefly given here in advance: will represent to us 
an empty domain 

Schroder then draws from his definition, the consequence that 
'nothing' is 'subject of every predicate' (46.04). 

B. NULL CLASS AND ASSERTORIC SYLLOGISTIC 

The concept of the null class leads once more to the positing of 
the scholastic problem of the validity of certain theorems in the 
assertoric syllogistic. On this point we have, exceptionally a tho- 
rough historical investigation in Albert Menne's excellent work 
(46.05). The discussion of this problem took, in brief, this course: 

Leibniz met with certain difficulties in his assertoric syllogistic: 
he was unable to deduce the four moods whose names contain 'p', 
which was the cause of his constantly building new systems without 
finding any of them satisfactory. Boole deduced all the moods apart 
from those four and the five subalternate ones, expressing the 
Aristotelian propositions in the fashion aforesaid (40.08). But he 
said nothing about the non-deducibility of the nine others. Venn 
and Schroder, however, went into the problem. Schroder says: 

46.06 From the stand-point of our theory we must now 
describe a number of these (syllogistic) moods as incorrect, 
viz. all those inferences by means of which a particular judg- 
ment is drawn from purely universal premisses. On closer 
inspection we shall see that these are enthymemes which tacitly 
omit an essential premiss - but as soon as this is explicitly 
formulated and added to complete the other premisses, then 
they evidently depend on three premisses and so cease to be 
simple' syllogisms, and even 'syllogisms' at all. 

The missing premiss is formulated thus: 

46.07 The inference only holds good when to the stated 
premisses the further, assumption a =|= is added as another 

366 



LOGIC OF GLASSES 

premiss, i.e. the supposition that there are individuals of the 
class of the subject. 

Hence two Aristotelian rules are rejected: 

46.08 Hence it is to be noted: that an inference by sub- 
alternation is not permissible in exact logic. 

46.09 It is . . . further to be noted: Of the conversions in 
traditional logic only the conversio pura is permissible in exact 
logic (i.e. not the conversio per accidens: cf. 32.08). 

The moods in question here are: Darapti, Felapton, Bamalip, 
Fesapo and the five subalternate moods: Barbari, Celaront, Cesaro, 
Camestrop, Calemop. 

It may be asked whether the interpretation given by Boole and 
Schroder to the Aristotelian propositions is the only possible one? 
Further development has shown that it is not. If 'all . . . are . . .' 
and 'some . . . are . . .' are taken as undefined symbols, a correct 
system can be constructed in which all the Aristotelian moods are 
valid. The following axioms, due to Lukasiewicz, are sufficient - 
apart from metalogical suppositions - to yield such a system. We 
translate them into ordinary language. 

1. All As are A 

2. Some As are A 

3. Barbara 

4. Dalisi 

By introducing a term-negation, one can even limit oneself to 1,2 
and 4, as I. Thomas has shown (46.10). The Aristotelian system has 
also been developed as an exact, demonstrably consistent (46.11) 
system, in which the traditional syllogisms are to hand as simple 
syllogisms - contrary to the statement of Schroder cited above. It 
is true that the system involves some further presuppositions, 
but this is so of every system of term-logic, not excluding Schroder's. 

C. DESCRIPTION 

1. The definite article: Frege 

Along with the problem of the null class, there has been posed 
in mathematical logic also that of the question: 'what is it that 
properly exists?' Frege was the first to give an answer to this, to 
the effect that existence is a property of the concept, not of the object 
(39.11-12). On the other hand, Frege introduced the concept of 
description (corresponding to the singular definite article), in depen- 
dence on his ideas about definitons of classes (45.02). His most 
important text on this subject is ithe following: 

46.13 If we allowed ourselves to assert as universally valid 
the equation of 'e (A = e)' with 'A', we should have in the 

367 



MATHEMATICAL VARIETY OF LOGIC 

form 'e <D (s)' a substitute for the definite article in language. 
For if it were supposed that (£) was a concept under which 
that object A - and only this - fell, — \^j — (a) = (A = a) 
would be the True, and so s (s) = k (A = s) would also be 
the True, and consequently on our equation of 'e (A = e)' 
with 'A', e (s) would be the same as A; i.e. in the case that 
(£) is a concept under which one and only one object 
falls, 'e (e), denotes this object. But this is of course not 
possible, since that equation, in its universality, must be 
admitted to fail. However, we can get some assistance by 
introducing the function 

with the stipulation that two cases are to be distinguished : 

1) if there corresponds to the argument an object A, in 
such a way that s (A = s) is the argument, the value of the 
function \ \ is A itself; 

2) if there corresponds to the argument no object A, in 
such a way that s (A = e) is the argument, the argument 
itself is the value of the function \ £. 

Accordingly \ s (A = s) = A is the True, and \ e (s)' stands 
for the object falling under the concept (5), when (5) is a 
concept under which one and only one object falls; in all 
other cases \ s (e)' stands for the same thing as l k O (s)\ 
Thus, e.g., 2 = \ I (s + 3 = 5) is the true, since 2 is the only 
object falling under the concept 

what added to 3 yields 5 
- presupposing a proper definition of the sign of addition - . . . 

Here we have a substitute for the definite article in speech, 
which serves to form proper names from conceptual phrases. 
E.g. from the word 

'positive square root of 2' 
which stand for a concept, we form the proper name 
'the positive square root of 2'. 

There is a logical danger here. For if we want to form the 
proper name 'the square root of 2' from the words 'square root 
of 2', we make a logical mistake, since this proper name 
would be ambiguous and even lacking in reference without 
further stipulations. If there were no irrational numbers, 
which has indeed been maintained, the proper name 'the 
positive square root of 2' would be lacking in reference, 
at least according to the immediate sense of the words, 
without further stipulations. And if we purposely assigned a 

368 



LOGIC OF CLASSES 

reference to this proper name, this would have no connection 
with its formation, and it could not be inferred that there is a 
positive square root of 2, though we should be only too ready 
to infer this. This danger concerning the definite article is 
here quite removed, since \ k O (e)' always has a reference, 
whether the function O (£) is not a concept, or is a concept 
under which more than one or no object falls, or whether 
again it is a concept under which one and only one concept 
falls. 

2. Logical existence 

Frege's theory was adopted by Russell in the 20th century, and 
further developed. This came about in peculiar circumstances. 
Russell himself, in 1901, introduced the distinction between real 
and logical existence: 

46.14 Numbers, the Homeric gods, relations, chimeras and 
four-dimensional spaces all have being, for if they were not 
entities of a kind, we could make no propositions about 
them. Thus being is a general attribute of everything, and to 
mention anything is to show that it is. 

Existence, on the contrary, is the prerogative of some only 
amongst beings. To exist is to have a specific relation to 
existence - a relation ... which existence itself does not have. 

The distinction is not very profound - compared with the Thomi- 
stic theory of ens ralionis (26.04 ff.) it seems incomplete: it confuses 
such different kinds of beings as relations, mathematical structures 
and fictitious heroes. But the text is interesting because two years 
later A. Meinong formulated very similar ideas which became the 
starting-point of Russell's theory of description. We cite a text 
from Meinong's famous Uber Annahmen. 

46.15 If anyone forms the judgment e.g. 'a perpeluum 
mobile does not exist', it is clear that the object of which 
existence (Dasein) is here denied, must have properties, and 
even characteristic properties, for without such the belief in 
non-existence can have neither sense nor justification; but 
the possession of properties is as much as to say a manner of 
being ( i soseiri > - Meinong's quotes). This manner of being, 
however, does not presuppose any existence, which is rather, 
and rightly, just what is denied. The same could be shown 
analogously about knowledge of components. By keeping in 
general, as has often been found helpful, to knowledge of, 

369 



MATHEMATICAL VARIETY OF LOGIC 

or the effort to know, how the object under consideration 
was conceived in two stages, the grasping of the object 
and the judging about it, it at once becomes evident that 
one may say : objects are grasped, so to speak, in their manner 
of being; what is then judged, and eventually assented to, 
is the being, or a further manner of being, of what is grasped 
in that manner of being. This manner of being, and through 
it that which is in this manner, is comprehensible without 
limitation to existence, as the fact of negative judgments 
shows ; but to that extent our comprehension finds something 
given about the objects, without respect to how the question 
of existence or non-existence is decided. In this sense 'there 
are' also objects which do not exist, and I have expressed this 
in a phrase which, while somewhat barbarous, as I fear, is 
hard to better, as 'externality (Aussersein) of the pure 
object'. 

A year later Meinong had extended this doctrine also to 'impos- 
sible' objects: 

46.16 There is, then, no doubt that what is to be an object 
of knowledge does not in any way have to exist. . . . The fact 
is of sufficient importance for it to be formulated as the 
principle of the independence of manner of being from exi- 
stence, and the domain in which this principle is valid can best 
be seen by reference to the circumstance that there are 
subject to this principle not only objects which in fact do not 
exist, but also such as cannot exist because they are impossible. 
Not only is the oft-quoted golden mountain golden, but the 
round square too is as surely round as it is square. ... To 
know that there are no round squares, I have to pass judgment 
on the round square. . . . Those who like paradoxical expres- 
sions, can therefore say: there are objects of which it is true 
that there are no objects of that kind. 

Nobody will deny that Meinong's doctrine is certainly paradoxi- 
cal. But it is also simply false: it is not necessary to pass judgment 
on a round square in order to know that there are no round squares. 
That a philosopher of Meinong's quality could commit so grave - and 
so perilous - an error, is due to his not conducting an exact logical 
analysis of the matters at issue, i.e. more precisely, that he was not 
acquainted with Frege's doctrine of description. This was first 
brought into notice by Russell. 

370 



LOGIC OF CLASSES 

3. Description in Russell 

It should be obvious that Russell's earlier theory (46.14) coincides 
with that of Meinong; though Russell goes further than Meinong 
in ascribing to Homeric gods etc. not merely a special manner of 
being but simply being. However, it seems that Meinong's clear 
formulation induced Russell to reject this doctrine and therewith 
his own earlier theory. He took over instead the theory of Frege and 
proceeded to develop it. In his paper On Denoting (1905) he wrote: 

46.17 The evidence for the above theory is derived from the 
difficulties which seem unavoidable if we regard denoting 
phrases as standing for genuine constituents of the propo- 
sitions in whose verbal expressions they occur. Of the pos- 
sible theories which admit such constituents the simplest is 
that of Meinong. This theory regards any grammatically 
correct denoting phrase as standing for an object. Thus 'the 
present King of France', 'the round square', etc. are supposed 
to be genuine objects. It is admitted that such objects do not 
subsist, but, nevertheless, they are supposed to be objects. This 
is in itself a difficult view; but the chief objection is that such 
objects, admittedly, are apt to infringe the law of contradiction. 
It is contended, for example, that the existent present King 
of France exists, and also does not exist; that the round 
square is round, and also not round, etc. But this is intolerable ; 
and if any theory can be found to avoid this result, it is 
surely to be preferred. 

The above breach of the law of contradiction is avoided by 
Frege's theory. . . . 

The theory of description is now introduced as follows, where 
Russell begins with an analysis of description : 

46.18 By a 'denoting phrase' I mean a phrase such as any 
one of the following; a man, some man, any man, every man, 
all men, the present King of England, the present King of 
France, the centre of mass of the solar system at the first 
instant of the twentieth century, the revolution of the earth 
round the sun, the revolution of the sun round the earth. 
Thus a phrase is denoting solely in virtue of its form. We may 
distinguish three cases: (1) A phrase may be denoting, and 
yet not denote anything; e.g., 'the present King of France'. 
(2) A phrase may denote one definite object; e.e. 'the present 
King of England' denotes a certain man. (3) A phrase may 
denote ambiguously; e.g. 'a man' denotes not many men, 

371 



MATHEMATICAL VARIETY OF LOGIC 

but an ambiguous man. The interpretation of such phrases is a 
matter of considerable difficulty. . . . 

46.19 My theory, briefly, is as follows. I take the notion 
of the variable as fundamental; I use l C(x)' to mean a propo- 
sition (Footnote: More exactly a propositional function.) . . . 
where x, the variable, is essentially and wholly undetermined. 
Then we can consider the two notions l C(x) is always true' 
and *C(x) is sometimes true'. Then everything and nothing and 
something (which are the most primitive of denoting phrases) 
are to be interpreted as follows : 

C(everything) means l C(x) is always true' ; 

C7(nothing) means iil C(x) is false" is always true'; 

C(something) means 'It is false that "C(x) is false" is 

always true'. 
Here the notion l C(x) is always true' is taken as ultimate and 
indefinable, and the others are defined by means of it. Every- 
thing, nothing, and something are not assumed to have any 
meaning in isolation, but a meaning is assigned to every 
proposition in which they occur. . . . 

Next Russell interprets the proposition, 'I met a man' as '"I 
met x, and x is human" is not always false', and sets out the theory 
of formal implication already described (44.11 ff.). There follows 
the theory of definite description : 

46.20 It remains to interpret phrases containing the. 
These are by far the most interesting and difficult of denoting 
phrases . . . the, when it is strictly used, involves uniqueness. . . . 
Thus when we say l x was the father of Charles II' we not only 
assert that x had a certain relation to Charles II, but also that 
nothing else has this relation. The relation in question, 
without the assumption of uniqueness, and without any 
denoting phrases, is expressed by l x begot Charles II'. To get 
an equivalent of 'x was the father of Charles II' we must add, 
'If y is other than x, y did not beget Charles II', or, what is 
equivalent, 'If y begot Charles II, y is identical with x\ Hence 
'# is the father of Charles IT becomes: 'x begot Charles II; 
and "if y begot Charles II, y is identical with x" is always 
true of y\ 

46.21 The whole realm of non-entities, such as 'the round 
square', 'the even prime other than 2', 'Apollo', 'Hamlet', 
etc., can now be satisfactorily dealt with. All these are denot- 
ing phrases which do not denote anything. . . . So . . . 'the round 

372 



LOGIC OF CLASSES 

square is round' means 'there is one and only one entity x 
which is round and square, and that entity is round', which is 
a false proposition, not, as Meinong maintains, a true one. 
'The most perfect Being has all perfections; existence is a 
perfection; therefore the most perfect Being exists' becomes: 
'There is one and only one entity x which is most perfect; 
that one has all perfections; existence is a perfection; therefore 
that one exists'. As a proof, this fails for want of a proof of the 
premiss 'there is one and only one entity x which is most 
perfect'. 

4. Symbolism 

a. Peano 

It only remained to introduce a suitable symbolism. This had 
already been created by Peano in connection with that used to 
define classes, and so with the plural article (cf. 45.06) : 

46.22 Let p be a P (i.e. proposition) containing a letter x; 
the formula x 3 p represents the class of xs which satisfy the 
condition p. 

The sign 3 can be read as the word 'which'. . . . 

Let us call the class x 3 p, a; the proposition x e a coincides 
with p; then every P containing a letter x, that is to say 
every condition in x, is reducible to the form x s a, where a is 
a determinate Cls (i.e. class). 

We also have x 3 [x e a) = a, x s (x 3 p) = p ; the two signs, 
x s and x 3 represent inverse operations. 

46.23 Let a be a class: a signifies: 'there are as, as exist'. 

46.24 i x = y 2 (y = x) {= (equal to x)} Df 
yzix. = .yz(ix):aDix. = a d (i x): 

a = ix, = .a = (ix) Df 

. . . This sign i is the first letter of the word foo?. So i x desig- 
nates the class formed by the object x, and i x w i y the class 
composed of the objects x and y. 

46.25 a s Cls. 3 a : x, y s a . d x,y, . x = y : = y: d : z = i a. 
= . a = i z . . . 

a e Cls ,a = ix.D.x = ix... 

Let a be a class containing a sole individual x. That is the 
case when there are as and two individuals of the class a are 
necessarily equal. In that case ? a . . . wdiich can be read 'the a' 
indicates the individual x which forms the class a. 

373 



MATHEMATICAL VARIETY OF LOGIC 

b. Principia 

The most important definitions in this connection in the Principia 
are these: 

46.26 *24.01. V = x (x =x) Df . . . 
*24.02. A = - V Df 

'V corresponds to Boole's '1', 'A' to his '0'. 

46.27 *24.03. 3 ! a . = . (3 x) . x s a Df 

46.28 *14.01. [( ix) (<p x) . <|i (? x )(? <r) . = : (3 b) : yx . =* 
.x = b\\b Df 

Here '(ix) (9 a;)' is an 'incomplete symbol' and is to be read 'the x 
such that <p#'; ty (1 x) (cp#)' ascribes the property ty to the x so 
described. The whole * 14.01 is to mean: there is at least one b such 
that tyb (at the end), and for ail x: <px if and only if x = b. 

46.29 *14.02. E ! (7 x) (9 sc) . = : (3 b) : 9 x . = > . x = b 
Df 

To say that the x, such that cpa?, exists, means to say that there is 
just one x such that qxr, i.e. one and only one such. 



374 



V. OTHER DOCTRINES 



§47. LOGIC OF RELATIONS 

The formal logic of relations is one of the chief new creation- of 
mathematical logic. Anticipations of it are, indeed, to be found in 
antiquity (Aristotle, 16.20ff. : Galen, 24.36) and among the Scho- 
lastics (cf. 35.12), but there is no developed theory before Lambert 
in the 18th century. Here we show the development from 1847, 
beginning with the basic doctrines in De Morgan (1847), and going 
on to Peirce (1883), and Russell (1903 and 1910). We close with some 
texts on the ancestral relation and isomorphy. 

A. LAYING THE FOUNDATIONS 
1. De Morgan 

The real founder of the modern logic of relations is De Morgan' 
of whom Peirce, himself a great logician, said that he 'was one of 
the best logicians that ever lived and unquestionably the father 
of the logic of relatives' (47.01). 

One pioneer text of De Morgan's has already been quoted (42.01) ; 
the following series comes from a paper of 1860: 

47.02 I now proceed to consider the formal laws of relation, 
so far as is necessary for the treatment of the syllogism. Let 
the names AT, Y, Z, be singular: not only will this be sufficient 
when class is considered as a unit, but it will be easy to 
extend conclusions to quantified propositions. 

47.03 Let X. .LY signify that X is some one of the objects 
of thought which stand to Y in the relation L, or is one of the 
Ls of Y. Let X. LY signify that X is not any one of the Ls 
of Y. Here X and Y are subject and predicate: these names 
having reference to the mode of entrance in the relation, not to 
order of mention. Thus Y is the predicate in LY . X, as well 
as in AT. LF. 

This is certainly a remarkable extension of the concept of subject 
and predicate. De Morgan's successors did not adopt it. 

47.04 When the predicate is itself the subject of a relation, 
there may be a composition: thus if X . . L{MY), if A' be one 
of the Ls of one of the Ms of Y, we may think of A' as an 
'L of M' of Y, expressed by X . . (LM)Y 1 or simplv by 
X . . LMY. 

375 



MATHEMATICAL VARIETY OF LOGIC 
De Morgan has thus introduced the concept of the relative product. 

47.05 We cannot proceed further without attention to 
forms in which universal quantity is an inherent part of the 
compound relation, as belonging to the notion of the relation 
itself, intelligible in the compound, unintelligible in the 
separated component. 

47.06 We have thus three symbols of compound relation; 
LM, an L of an M; LM', an L of every M; L, M, an L of none 
but Ms. No other compounds will be needed in syllogism, 
until the premises themselves contain compound relations. 

47.07 The converse relation of L, L _1 , is defined as usual: 
HX ..LY, Y..L- 1 : if X be one of the Ls of F, F is one of the 
L x s of X. And L 1 may be read 'L-verse of X\ Those who 
dislike the mathematical symbol in L 1 might write L x . This 
language would be very convenient in mathematics : cp- 1 ^ 
might be the '<p-verse of x' read as '<p-verse x\ 

Relations are assumed to exist between any two terms 
whatsoever. If X be not any L of Y, X is to Y in some not-L 
relation : let this contrary relation be signified by /; thus X . LY 
gives and is given by X . . IY. Contrary relations may be 
compounded, though contrary terms cannot: Xx, both X 
and not-X, is impossible; but Llx, the L of a not-L of X, is 
conceivable. Thus a man may be the partisan of a non-partisan 
of X. 

47.08 Contraries of converses are converses : thus not-L 
and not-L- 1 are converses. For X . . LY and Y . . L X X are 
identical; whence X . . not-L Fand F . . (not-L- 1 )^, their 
simple denials, are identical; whence not-L and not-L- 1 are 
converses. 

Converses of contraries are contraries: thus L 1 and (not- 
L) 1 are contraries. For since X . . LY and X . . not-L F are 
simple denials of each other, so are their converses F . . L X X 
and F . . (not-L) ml X\ whence L 1 and (not-L)- 1 are contraries. 

The contrary of a converse is the converse of the contrary: 
not-L 1 is (not-L)- 1 . For X . . LY is identical with F. not- 
L*X and with X . (not-L) F, which is also identical with F. 
(not-L) X X. Hence the term not-L-verse is unambiguous in 
meaning though ambiguous in form. 

If a first relation be contained in a second, then the converse 
of the first is contained in the converse of the second : but the 
contrary of the second in the contrary of the first. 

376 



LOGIC OF RELATIONS 

The conversion of a compound relation converts both 
components, and inverts their order. 

47.09 A relation is transitive when a relative of a relative is 
a relative of the same kind; as symbolized in LL))L, whence 
LLL))LL))L; and so on. 

A transitive relation has a transitive converse, but not 
necessarily a transitive contrary; for L l L l is the converse of 
LL, so that LL))L gives L^L^LK 

2. Peirce 

47.10 A dual relative term, such as 'lover', 'benefactor', 
'servant', is a common name signifying a pair of objects. Of 
the two members of the pair, a determinate one is generally 
the first, and the other the second; so that if the order is 
reversed, the pair is not considered as remaining the same. 

Let A, B, C, D, etc., be all the individual objects in the 
universe; then all the individual pairs may be arrayed in a 



block, thus: 








A : A 


A : B 


A : C 


A : D etc. 


B : A 


B : B 


B: C 


B : D etc. 


C :A 


C : B 


C : C 


C : D etc. 


D :A 


D : B 


D : C 


D : D etc. 


etc. 


etc. 


etc. 


etc. etc. 



A general relative may be conceived as a logical aggregate 
of a number of such individual relatives. Let / denote 'lover'; 
then we may write 

/ = SiSy (/)</(/: J) _ 
where (/)// is a numerical coefficient, whose value is 1 in case 
/ is a lover of J, and in the opposite case, and where the 
sums are to be taken for all individuals in the universe. 

Peirce therefore takes relations extensionally, as classes of 
pairs. 

47.11 Every negative term has a negative (like any other 
term) which may be represented by drawing a straight line 
over the sign for the relative itself. The negative of a relative 
includes every pair that the latter excludes, and vice versa. 
Every relative has also a converse, produced by reversing the 
order of the members of the pair. Thus, the converse of "lover" 
is "loved". The converse may be represented by drawing a 
curved line over the sign for the relative, thus: /. It is defined 
by the equation 



MATHEMATICAL VARIETY OF LOGIC 

(l)'i = (/)/(• 

The following formulae are obvious, but important: 
1 = 1 1=1 

1 = 1 
(*■_< 6) = (5 -< J) (l^b) = (l-^b). 

Relative terms can be aggregated and compounded like 
others. Using + for the sign of logical aggregation, and the 
comma for the sign of logical composition (Boole's multip- 
lication, here to be called non-relative or internal multip- 
lication), we have the definitions 

(l + b)u = (l)ij + (b)ij 
(I fi)u = (l)ij x (b)ij. 
The first of these equations, however, is to be understood in 
a peculiar way: namely, the + in the second member is not 
strictly addition, but an operation by which 

+ = + 1 = 1+0 = 1+1 = 1. 

That is to say that Peirce, unlike Boole, uses non-exclusive 
disjunction (40.11). 

47.12 The main formulae of aggregation and composition 
are 

If / — < s and b — < s, then / + b — < s. 
If s — < / and s — < b, then s — < /, b. 
If / + b — < s, then / — < s and b — < s. 
If s — < /, 6, then s — < / and s — < b. 
J (/ + b), s — < /, s + b, s. 
\( I + s), (b + s) — < /, b + s. 
The subsidiary formulae need not be given, being the same as 
in non-relative logic. 

47.13 We come now to the combination of relatives. Of 
these, we denote two by special symbols; namely, we write 

lb for lover of a benefactor, 
and 

/ f b for lover of everything but benefactors. 
The former is called a particular combination because it 
implies the existence of something loved by its relate and a 
benefactor of its correlate. The second combination is said to 
be universal, because it implies the non-existence of anything 
except what is either loved by its relate or a benefactor of its 
correlate. 

In the first case, (lb), we have, as can be seen from the formula 
given below (47.14), a situation like this: x (relate) loves y, and y is 

378 



LOGIC OF RELATIONS 

a benefactor of z (correlate); so the relation holds between x and z, 
provided always that there is (at least) one y which is loved by x 
and is a benefactor of z. 

tklAk The combination lb is called a relative product, / f b 
a relative sum. The / and b are said to be undistributed in 
both, because if / — < s, then lb — < sb and / f b — < % f b; and 
if 6 — < s, then /6 — < Is and / f 6 — < / f s. 

The two combinations are defined by the equations 

(lb)ij = 2* (l)ix (b)xj 

(/ t b)u = Ux {(/)/* + (b)xj) 
The sign of addition in the last formula has the same sig- 
nification as in the equation defining non-relative multipli- 
cation. 

Relative addition and multiplication are subject to the 
associative law. That is, 

lt(*t«) = (*t&)t«. 

/ {bs) = (lb) s. 

Two formulae so constantly used that hardly anything can 
be done without them are 

I [b t *) — < lb t s, 

(/ f 6) s — < / f &s. 
The former asserts that whatever is lover of an object that is 
benefactor of everything but a servant, stands to everything 
but servants in the relation of lover of a benefactor. The latter 
asserts that whatever stands to any servant in the relation of 
lover of everything but its benefactors, is a lover of everything 
but benefactors of servants. The following formulae are 
obvious and trivial: 

Is + bs — < (I + b) s 

f, 6 t > — < c t •), (& t •)■ 

Unobvious and important, however, are these: 

(/ + b) s — < Is + bs 

(/ 1 «),(* t •)-<'. * t*. 

There are a number of curious development formulae. . . . 

We pass over Frege, Peano and Schroder - a text of Frege's 
will be given later (47.30 ff.) - and go on at once to Russell. 

3. Russell 

Russell strangely seems at first to have adopted an intensional 
view of relations. 

47.15 Peirce and Schroder have realized the great impor- 

379 



MATHEMATICAL VARIETY OF LOGIC 

tance of the subject (i.e. of the logic of relations), but unfor- 
tunately their methods, being based, not on Peano, but on the 
older Symbolic Logic derived (with modifications) from 
Boole, are so cumbrous and difficult that most of the appli- 
cations which ought to be made are practically not feasible. In 
addition to the defects of the old Symbolic Logic, their 
method suffers technically (whether philosophically or not I 
do not at present discuss) from the fact that they regard a 
relation essentially as a class of couples, thus requiring 
elaborate formulae of summation for dealing with single 
relations. This view is derived, I think, probably uncons- 
ciously, from a philosophical error: it has always been custo- 
mary to suppose relational propositions less ultimate than 
class-propositional (or subject-predicate propositions, with 
which class-propositions are habitually confounded), and this 
has led to a desire to treat relations as a kind of classes. 
However this may be, it was certainly from the opposite 
philosophical belief, which I derived from my friend Mr. G. E. 
Moore, that I was led to a different formal treatment of 
relations. This treatment, whether more philosophically 
correct or not, is certainly far more convenient and far more 
powerful as an engine of discovery in actual mathematics. 

Certainly experience has shown clearly that the extensional 
view is more convenient. We read in the Principia: 

4. Principia 

47.16 A relation, as we shall use the word, will be under- 
stood in extension: it may be regarded as the class of couples 
(x, y) for which some given function <\> (x, y) is true. 

47.17 The following is the definition of the class of rela- 
tions : 

*21.03. Rel = ft {{3 9) . R = xy 9 ! {x, y)} Df 
Similar remarks apply to it as to the definition of 'Cls' ( *20.03) 
(cf. 45.07). . . . The notation 'xRy' will mean l x has the 
relation R to y\ This notation is practically convenient, and 
will, after the preliminaries, wholly replace the cumbrous 
notation x {xy 9 (x, y)} y. 

The most important basic notions of relation-logic in the Principia 
are these: 

47.18 (23.01. R cS.=: xRy . Dx, y ,.xSy Df 
380 



LOGIC OF RELATIONS 

*23.02 RnsS = xy (xRy . xSy) Df 

*23.03 R^S = xy (xRy . v . xSy) Df 

*23.04 - R = xy{ ~ (xRy) } Df 

47.19 The universal relation, denoted by V , is the relation 
which holds between any two terms whatever of the appro- 
priate types, whatever these may be in the given context. 
The null relation, A, is the relation which does not hold be- 
tween any pair of terms whatever, its type being fixedby the 
types of the terms concerning which the denial that it holds 
is significant. A relation R is said to exist when there is at least 
one pair of terms between which it holds; 'R exists' is written 
'# ! R\ . . . 

*25.01. V = xy(x = x.y =y) Df 

*25.02. A = -»■ V Df 

*25.03. 3 ! R . = . (3 x, g) . xRy Df 

47.20 The general definition of a descriptive function is: 
*30.01. Ry = (ix){xRy) Df 

That is, '.R'z/' is to mean 'the term x which has the relation 
R to y\ If there are several terms or none having the rela- 
tion R to y, all propositions about R'y, i.e. all propositions of 
the form '<p(i?'i/)', will be false. The apostrophe in 'R l y' may 
be read 'of. Thus if R is the relation of father to son, 'R'y' 
means 'the son of y' ; in this case, all propositions of the 
form l (p(R'yy will be false unless y has one son and no more 
(cf. 46.20). 

47.21 If R is a relation, the relation which y has to x when 
xRy is called the converse of R. Thus greater is the converse of 
less, before of after, husband of wife. The converse of identity 
is identity, and the converse of diversity is diversity. The 
converse of R is written R (read '/?-converse'). When R = 
R, R is called a symmetrical relation, otherwise it is called 
not-symmetrical. When R is incompatible with R, R is called 
asymmetrical. Thus 'cousin' is symmetrical, 'brother' is not- 
symmetrical (because when x is the brother of y, y may be 
either the brother or the sister of sc), and 'husband' is asym- 
metrical. 

47.22 Given any relation R, the class of terms which have 
the relation R to a given term y are called the referents of y, 
and the class of terms to which a given term x has the relation 

R are called the relata of x. We shall denote by R the relation 

of the class of referents of y to y, and by R the relation of the 

381 



MATHEMATICAL VARIETY OF LOGIC 

class of relata of x to x. . . . R and R are chiefly useful for the 
same of the descriptive functions to which they give rise ; thus 

R'y = x (xRy) and R l x = y (xRy). Thus e.g. if R is the rela- 
tion of parent to son, R l y the parents of y, R'x the sons of x. 
If R is the relation of less to greater among numbers of any 

kind, R'y = numbers less than y, and R'x = numbers greater 

than x. When R'y exists, R'y is the class whose only member 
is R'y. But when there are many terms having the relation R 

to y, R'y, which is the class of those terms, supplies a notation 
which cannot be supplied by R'y. And similarly if there are 

many terms to which x has the relation R, R l x supplies the 
notation for these terms. Thus for example let R be the rela- 
tion 'sin', i.e. the relation which x has to y when x = sin y. 
Then 'sinV represents all values of y such that x = sin y, 
i.e. all values of sin 1 x or arcsin x. Unlike the usual symbol, 
it is not ambiguous, since instead of representing some one 
of these values, it represents the class of them. 

The definitions of R, R, . . . are as follows : 

*32.01. R = Sty { a = x {xRy) } Df 

*32.02. R = @ x {$=y {xRy) } Df 

47.23 If R is any relation, the domain of R, which we denote 
by D'R, is the class of terms which have the relation R to 
something or other; the converse domain, G'i?, is the class of 
terms to which something or other has the relation R; and the 
field, C l R, is the sum of the domain and the converse domain. 
(Note that the field is only significant when R is a homogeneous 
relation.) 

The above notations D'i?, Q'R, CR are derivative from 
the notations .D,G, C for the relations, to a relation, of its 
domain, converse domain, and field respectively. We are to 
have 

D'R = x{(3y).xRy} 
a i R=y{(3x).xRy} 
C i R = x{(3y):xRy.v .yRx}; 
hence we define D, a, C as follows: 
*33.01. D = dft[x = x{{3y).xRy}] Df 

*33.02. a = |8A[p =y{(3x).xRy}] Df 

*33.03. C = yfl [y = x{ (3 y) : xRy . v . yRx }] Df 
The letter C is chosen as the initial of the word "campus". 



382 



LOGIC OF RELATIONS 

47.24 The relative product of two relations R and S is the 
relation which holds between x and z when there is an inter- 
mediate term y such that x has the relation R to y and y has 
the relation S to z. Thus e.g. the relative product of brother 
and father is paternal uncle; the relative product of father 
and father is paternal grandfather; and so on. The relative 
product of R and S is denoted by "R | S"; the definition is: 
*34.01. R | S = xz{ (3 y) . xRy . ySz } Df . . . 

The relative product of R and R is called the square of R; 
we put 

*34.02. R 2 = R | /? Df 

*34.03. R 3 = i? 2 1 1? Df 

47.25 We have to consider the relation derived from a 
given relation i? by limiting either its domain or its converse 
domain to members of some assigned class. A relation R with 
its domain limited to members of a is written "a^ /?"; with its 
converse domain limited to members of p, it is written "i? [ p" ; 
with both limitations, it is written "a "| R [ (3". Thus e.g. 
"brother" and "sister" express the same relation (that of a 
common parentage), with the domain limited in the first 
case to males, in the second to females. "The relation of white 
employers to coloured employees" is a relation limited both as 
to its domain and as to its converse domain. We put 
*35.01. a 1 R = xy (x e a . xRy) Df 

with similar definitions for R [ a and a^| R [ (3. 

46.26 P I a is defined as follows : 
*36.01. P O = a1 P pa Df 
We thus have 

*36.13. h : x (P I a) y . = . x, y e ac , xPy 

47.27 We introduce what may be regarded as the plural 
of R l y. "R'y" was defined to mean "the term which has the 
relation R to y". We now introduce the notation "/?"£" to 
mean 'the terms which have the relation R to members of 
P". Thus if p is the class of great men, and i? is the relation of 
wife to husband, i?"(3 will mean "wives of great men". If (3 
is the class of fractions of the form 1-1/2° for integral values 
of n, and R is the relation "less than", R il $ will be the class of 
fractions each of which is less than some member of this 
class of fractions, i.e. /?"(3 will be the class of proper fractions. 
Generally, i?"(3 is the class of those referents which have 
relata that are members of (3. 

We require also a notation for the relation of i?"p to p. 

383 



MATHEMATICAL VARIETY OF LOGIC 

This relation we will call R e . Thus R 3 is the relation which 
holds between two classes a and (3 when a consists of all terms 
which have the relation R to some member of (3. 

A specially important case arises when Ry always exists if 
y e p. In this case, R li fi is the class of all terms of the form 
Ry when y e p. We will denote the hypothesis that R l y 
always exists if y s p by the notation EM i?"p, meaning "the 
Rs of (3's exist". 

The definitions are as follows: 
*37.01. i?"p = x{ (3 y) . y s (3 . xRy } Df 

*37.02. i?e = fci8(a = iT(3) Df 

47.28 A one-many relation is a relation R such that, if y 
is any member of G'f?, there is one, and only one, term x 

which has the relation R to y, i.e. R'y e 1. Thus the relation 
of father to son is one-many, because every son has one 
father and no more. The relation of husband to wife is one- 
many except in countries which practise polyandry. (It is 
one-many in monogamous as well as in polygamous countries, 
because, according to the definition, nothing is fixed as to the 
number of relata for a given referent, and there may be only 
one relatum for each given referent without the relation 
ceasing to be one-many according to the definition.) The 
relation in algebra of x 2 to x is one-many, but that of x to x 2 
is not, because there are two different values of x that give 
the same value of x 2 . 

47.29 A relation R is called many-one when, if x is any 
member of D'R, there is one, and only one, term y to which x 

has the relation R, i..e Rx e 1. Thus many-one relations are 
the converses of one-many relations. When a relation R is 
many-one, Rx exists whenever x e D'i?. 

A relation is called one-one when it is both one-many and 
many-one, or, what comes to the same, when both it and its 
converse are one-many. 

B. SERIES 

One of the most important parts ot the logic of relations is the 
doctrine of series of relations, which plays a notable part in mathe- 
matics and other sciences (e.g. Biology). It is based on the theory 
of the relative product (47.13 f.), and makes use of the difficult 
concept of the ancestral relation, which Frege was the first to 
define exactly. We give first some texts from the Begriffsschrift, 
then the elaboration of the concept in the Principia. 

384 



LOGIC OF RELATIONS 

1. Frege 

47.31 If from the proposition that b has a property F, it 
can be universally, whatever b may be, concluded that every 
result of the application of a process / to b has the property 
F, then I say: 'the property F is hereditary in the /-series'. 

47.32 If the property F is hereditary in the /-series : if x has 
the property F and y is the result of applying the process / 
to x: then y has the property F. 

47.33 If from the two propositions, that every result of 
applying the process / to x has the property F, and that the 
property F is hereditary in the /-series, whatever F may be, 
it can be concluded that y has the property F, then I say: 
l y succeeds x in the /-series'. 

47.34 If x has the property F which is hereditary in the 
/-series, and if y succeeds x in the /-series, then y has the 
property F. 

47.35 If y succeeds x in the /-series, and if z succeeds y in 
the /-series, then z succeeds x in the /-series. 

47.36 If from the circumstance that c is a result of applying 
the process / to b, whatever b may be, it can be concluded 
that every result of applying /to b is the same as e, then I 
say: 'the process / is unequivocal'. 

47.37 If x is a result of applying the unequivocal process / 
to y, then every result of applying the process / to y belongs 
to the /-series that begins with x. 

2. Principia 

47.38 Mathematical induction is, in fact, the application to 
the number-series of a conception which is applicable to all 
relations, and is often very important. The conception in 
question is that which we shall call the ancestral relation with 
respect to a given relation. If R is the given relation, w T e denote 
the corresponding ancestral relation by "R+"; the name is 
chosen because, if Ft is the relation of parent and child, R^ 
will be the relation of ancestor and descendant - where, for 
convenience of language, we include x among his own ancestors 
if # is a parent or a child of anything. 

It would commonly be said that a has to z the relation of 
ancestor to descendant if there are a certain number of 
intermediate people b, c, d, . . . such that in the series a, 6. c, 
d, . . . z each term has to the next the relation of parent and 

385 



MATHEMATICAL VARIETY OF LOGIC 

child. But this is not an adequate definition, because the dots 
in 

"a, b, c, d, . . . z" 
represent an unanalysed idea. 

47.39 Let us call \x a hereditary class with respect to R if 
jR" (xC[jl, i.e. if successors of [x's (with respect to R) are (x's. 
Thus, for example, if \i is the class of persons named Smith, 
[i is hereditary with respect to the relation of father to son. 
If [i is the Peerage, \i is hereditary with respect to the relation 
of father to surviving eldest son. If (x is numbers greater 
than 100, \i is hereditary with respect to the relation of v to 
v + 1 ; and so on. If now a is an ancestor of z, and \x is a 
hereditary class to which a belongs, then z also belongs to this 
class. Conversely, if z belongs to every hereditary class to 
which a belongs, then (in the sense in which a is one of his 
own ancestors if a is anybody's parent or child) a must be an 
ancestor of z. For to have a for one's ancestor is a hereditary 
property which belongs to a, and therefore, by hypothesis, 
to z. Hence a is an ancestor of z when, and only when, a 
belongs to the field of the relation in question and z belongs 
to every hereditary class to which a belongs. This property 
may be used to define the ancestral relation; i.e. since we have 

aR^z . = : a z C R : R" \l c [L . a e \l . d [l . z e [j. 
R^ = dz{a z C'R : R"[i c \i . a z \x . z>\x . z z \ij Df 

We then have 

h : a z C'R . d . R^ l a = z{R" \x c \i . a z \i . d\i . z z [l^. 

Here R^ l a may be called 'the descendants of a'. It is the class 
of terms of which a is an ancestor. 

c. ISOMORPHY 

Finally, with the help of relational concepts, another doctrine 
can be developed which is important for many sciences, viz. that of 
isomorphy or ordinal similarity. Essentially this is a matter of the 
identity of two formal structures, i.e. of two networks of relations, 
similar only in their purely formal properties, but in those identical. 
We have already met an anticipation of this theory in the Middle 
Ages (28.18 ff.). It is first presented in detail in the Principia. 
There too, the concept of isomorphy is applied to the theory of 
types, fundamental for the system, as a result of which we get the 
theory of so-called systematic ambiguity (48.23). 

47.40 Two series generated by the relations P and Q 
386 



ANTINOMIES AND TYPES 

respectively are said to be ordinally similar when their terms 
can be correlated as they stand, without change of order. 
P 

X y y 

— * P = S I Q | S 

s S 



S l x 



V 



In the accompanying figure, the relation S correlates the 
members of C l P and C l Q in such a way that if xPy, then 
[S'x) Q (S l y), and if zQw, then {S l z) P (S'w). It is evident that 
the journey from x to y (where xPy) may, in such a case, be 
taken by going first to S'x, thence to S'y, and thence back to y, 
so that xPy . = . x {S \ Q \ S) y, i.e. P = S\Q\ 8. Hence to 
say that P and Q are ordinally similar is equivalent to saying 
that there is a one-one relation 8 which has C l Q for its con- 
verse domain and gives P = S | Q \ S. In this case we call S 
a correlator of Q and P. 

We denote the relation of ordinal similarity by "smor", 
which is short for "similar ordinally". Thus 

Psmorg. =.(3S).Ssl ►! . CQ = d'S . P = S \Q\ S 



§ 48. ANTINOMIES AND THEORIES OF TYPES 

A. HISTORICAL SURVEY 

Former endeavours to solve the problem of semantic antinomies 
(§§ 23 and 35) seem to have fallen into complete oblivion in the 
time of the 'classical' decadence. Nor did the mathematical logicians 
know anything about them until Riistow (48.01), with the sole excep- 
tion of Peirce, who had read Paul of Venice and given a subtle com- 
mentary on him in this respect (48.02) ; but even Peirce seems to 
have attended only to one of the numerous solutions quoted above 
(§35). 

At the end of the 19th century the problem re-emerged, and in a 
new form. Besides the 'Liar' a whole series of antinomies are found 
which are not semantic but logical, i.e. which arise without the use 
of any metalinguistic expressions. And they are genuine antinomies, 
i.e. contradictions deducible from intuitively evident axioms by 
means of no less correct rules (which differentiates them from simple 
contradictions). In spite of the fact that the antinomies are clearly 



387 



MATHEMATICAL VARIETY OF LOGIC 

seen to be of this character, they are often called by the gentler 
name of 'paradoxes'. 

The history of the problem of antinomies in our time is, briefly, as 
follows. Between 1895 and 1897 C. Burali-Forti (48.03) and G. Can- 
tor (48.04) independently stated the first logical antinomy (of the 
set of all ordinal numbers). But logicians considered this antinomy 
to be a matter of mathematics in the narrow sense, and gave it 
little attention: people were already accustomed to the fact that 
what were ordinarily considered to be unassailable parts of mathe- 
matics had been in difficulties right from the time of Zeno of Elea. 
In 1902 B. Russell constructed his celebrated antinomy of the class 
of all classes, which was first published by Frege (48.05) who also 
proposed an improvement on it (48.06). Subsequently many fresh 
antinomies, some logical, some semantic, have appeared. Up to 
now more than a dozen genuinely different ones are known. 

Naturally, logicians began to seek solutions, as they had done in 
antiquity and in the Middle Ages. To begin with, two attempts 
were made: Russell's ramified theory of types (1908) and Zermelo's 
theory (48.07). The latter was markedly mathematical in character, 
and cannot be expounded here. In 1910 the Principia was con- 
structed according to the ramified theory. In 1921 L. Chwistek 
simplified this by his construction of the simple theory of types, 
which received further development and confirmation at the hands 
of P. Ramsey in 1926. Both assisted towards clarification by apply- 
ing the theory of types to expressions, whereas Russell had not 
determined its semantic character. This line of development was 
terminated by St. Lesniewski's theory of semantic levels (48.08). 
Parallel with this work on the theory of types, new experiments 
were constantly being made to simplify it, or find an alternative. 
The plan of this work does not allow us to go further into the more 
recent developments. We confine ourselves, accordingly, to the 
antinomies themselves, and the two forms of the theory of types. 
We shall further add illustrations of two doctrines closely connected 
with the ramified theory of types, viz. that of the axiom of reduci- 
bility and that of systematic ambiguity. 



B. THE ANTINOMIES 

We now give some texts in which antinomies are formulated; 
others can be found in Becker (48.04). Our present texts always 
speak of contradictions, not antinomies, as the two had not then 
been distinguished; but antinomies are always intended. 

We read in the Principia: 

48.09 We shall begin by an enumeration of some of the 
more important and illustrative of these contradictions, and 



388 



ANTINOMIES AND TYPES 

shall then show how they all embody vicious-circle fallacies, 
and are therefore all avoided by the theory of types. It will 
be noticed that these paradoxes do not relate exclusively 
to the ideas of number and quantity. Accordingly no solution 
can be adequate which seeks to explain them merely as the 
result of some illegitimate use of these ideas. The solution 
must be sought in some such scrutiny of fundamental logical 
ideas as has been attempted in the foregoing pages. 

48.10 (1) The oldest contradiction of the kind in question 
is the Epimenides. . . . The simplest form of this contradiction 
is afforded by the man who says "I am lying"; if he is lying, 
he is speaking the truth, and vice versa (cf. 23.10fT.). 

48.11 (2) Let w be the class of all those classes which are 
not members of themselves. Then, whatever class x may be, 
"x is a w" is equivalent to "x is not an x". Hence, giving to x 
the value w, "w is a w" is equivalent to "w is not a w'\ 

This is Russell's own famous antinomy of the class of all classes. 
It differs from those mentioned so far in that it contains no seman- 
tic expressions such as T say', T lie', '. . . is true' etc., no statements 
about statements. Antinomies of this new kind are called 'logical' 
to distinguish them from the semantic ones. 

48.12 (3) Let T be the relation which subsists between two 
relations R and S whenever R does not have the relation R 
to S. Then, whatever relations R and 5 may be, "R has the 
relation T to S" is equivalent to "R does not have the relation 
jR to S". Hence giving the value T to both R and S, l T has 
the relation T to T" is equivalent to "T does not have the 
relation T to T". 

48.13 (5) The number of syllables in the English names of 
finite integers tends to increase as the integers grow larger, 
and must gradually increase indefinitely, since only a finite 
number of names can be made with a given finite number of 
syllables. Hence the names of some integers must consist of 
at least nineteen syllables, and among these there must be a 
least. Hence "the least integer not nameable in fewer than 
nineteen syllables" must denote a definite integer; in fact, 
it denotes 111,777. But "'the least integer not nameable in 
fewer than nineteen syllables" is itself a name consisting of 
eighteen syllables; hence the least integer not nameable in 
fewer than nineteen syllables can be named in eighteen 
syllables, which is a contradiction. 

389 



MATHEMATICAL VARIETY OF LOGIC 

This is another semantic antinomy. In the same context Russell 
gives three others, those of Burali-Forti, Richard, and the least 
indefinable ordinal number. We give the second in Richard's own 
words: 

48.14 I am going to define a certain set (ensemble) of 
numbers which I shall call the set E, by means of the following 
considerations : 

Let us write all the arrangements of the twenty-six letters 
of the French alphabet (taken) two by two, arranging these 
arrangements in alphabetic order; then all the arrangements 
three by three, ranged in alphabetic order; then those four 
by four, etc. These arrangements may contain the same letter 
repeated several times; they are arrangements with repetition. 

Whatever whole number p may be, every arrangement 
of the twenty-six letters p by p will be found in this table, 
and as everything that can be written with a finite number 
of words is an arrangement of letters, everything that can be 
written will be found in the table of which we have just shown 
the manner of construction. 

As numbers are defined by means of words, and the latter 
by means of letters, some of these arrangements will be 
definitions of numbers. Let us cancel from our arrangements 
all those which are not definitions of numbers. 

Let u x be the first number defined by an arrangement, 
u 2 the second, u 3 the third, etc. 

There have thus been arranged in a determinate order all 
the numbers defined by means of a finite number of words. 

Therefore : all the numbers that can be defined by means of 
a finite number of words form a denumerable set. 

This now is where the contradiction lies. We can form a 
number which does not belong to this set. 

'Let p be the n-th decimal of the n-th number of the set E; 
let us form a number having zero for its integral part, p + 1 
for its Ti-th decimal if p is equal neither to eight nor to nine, 
and otherwise unity.' 

This number N does not belong to the set E. If it was the 
n-th number of the set E, its n-th figure would be the n-th 
decimal figure of that number, which it is not. 

I call G the group of letters in inverted commas. 

The number N is defined by the words of the group G, i.e. 
by a finite number of words; it ought therefore to belong to 
the set E. But we have seen that it does not belong. 

390 



ANTINOMIES AND TYPES 

That is the contradiction. 

Richard then tries to show that the contradiction is only apparent. 

C. ANTICIPATIONS OF THE THEORY OF TYPES 

Peano's distinction between e and d (15.12) can be regarded as 
already a beginning of the later theory of types. An idea of Schroder's 
which plays the same part in his system as that distinction in Peano's, 
is a much closer approach.* 

48.16 In the last example, the subsumption 42 1, it can 
further easily be shown that it is not in fact permissible to 
understand by 1 a class so extensive, as it were so completely 
open, as the 'universum of discussability' depicted above. 

As has been shown, is to be contained in every class which 
can be selected from the manifold, 1, so that £ a holds, 
is to be subject to every predicate. 

If now we were to understand by a, the class of those classes 
which are equal to 1, [and this would certainly be permissible, 
if we could include in 1 everything conceivable],* then this 
class comprises essentially only one object, viz. the symbol 1 
itself or the totality of the manifold which constitutes its 
reference - but beyond that also 'nothing', in view of 0. But if 1 
and constitute the class of those objects which can be 
equated with 1, not only: 1 = 1, but also: = 1, must be 
admitted. Since a predicate which belongs to a class [here the 
predicate: being identically equal to 1] must also belong to 
every individual of this class, .... 

In such a manifold, in which = 1 holds, all possibility of 
distinguishing two classes or even individuals, is antecedently 
excluded; everything would here be all the same ('wurst'). 

48.17 These considerations show, that Boole's universal 
interpretation of 1 was, in fact, too extreme. 

In the actual calculus of domains, the subsumption =£] a 
can, as we have seen, be held as valid without limitation, e.g. 
for the domain a for a manifold of points, 1. 

But now the question must be answered, how far the laws 
of the calculus can be carried over also to the manifold 
formed of all possible classes, of any objects of thought whatever. 

*This text of Schroder's was brought to our notice by a paper of Prof. A. Church 
(48.15). The author kindly made available this work which is obtainable only 
with difficulty. 

**The square brackets are Schroder's. 

391 



MATHEMATICAL VARIETY OF LOGIC 

It has been shown that it is not permissible to leave this 
manifold, 1, completely undetermined, wholly unlimited or 
open, since some possible formulations of the predicate- 
class a . . . prove not to be permissible. How then are we to 
secure that the rules of the calculus, when applied to it, . . . 
can no longer lead to contradiction? 

I shall try to give the answer to this difficult question. 

First of all, we are concerned with a manifold of any kind 
of 'things' - objects of thought in general - as 'elements' or 
'individuals'. These may be (in whole or in part) given from 
the start, or may be (in the other part or in whole) somehow 
determined only conceptually. But they cannot remain 
completely undetermined, as has already been shown. 

In order that the symbols and 1 etc. may be applicable 
according to the rules of the calculus in this manifold, certain 
stipulations must be made concerning how the elements of 
this manifold are given or determined conceptually. 

As a first stipulation we have already specified in § 7, in 
postulate ((1+))*, that: the elements of the manifold must be 
all of them consistent, 'compatible' with one another. Only in 
this case do we symbolize the manifold by 1. 

48.18 If the elements of the manifold are consistent, they 
can be arbitrarily collected together into systems, 'domains' 
of its elements, and distinguished within it. In other words, 
any classes of individuals can be selected from the manifold 
even for the purpose of distributive application. . . . 

By that process of arbitrary selection of classes of individuals 
of the manifold originally envisaged, there [in general] arises, 
is produced, a new, much more extensive manifold, viz. that 
of the domains or classes of the previous one. . . . 

The new manifold could be called the 'second power' of the 
former - but better its 'first . . . derived manifold'. 

From that in turn a new, still more extensive manifold 
can be derived, which is to be called the derivate of the first 
derivate, or the second derivate. And so on. 

As can be seen from the foregoing considerations, the 
meaning of the identical 1 in the first manifold cannot extend 
to that on the second, still less to that in still higher derived 
manifolds. 

And in order that the subsumption (2+) can be held as 

*The parentheses are Schroder's. 

392 



ANTINOMIES AND TYPES 

valid even in the original manifold, it is necessary [and 
sufficient] from the start that among the elements given as 
'individuals' there should be no classes comprising as elements 
individuals of the same manifold. 

It is remarkable that Frege did not perceive the significance of 
this doctrine. In his review of Schroder's Vorlesungen (48 '19), he 
even strongly opposed it. We give Frege's text, since it is so for- 
mulated as almost to lead one to suppose that he is attacking 
the simple theory of types. 

48.20 Herr Schroder derives from this the conclusion that 
the original manifold 1 must be so made up that, among the 
elements given as individuals within it, there are found no 
classes that, for their part, contain within themselves as 
individuals any elements of the same manifold. This expedient, 
as it were, belatedly gets the ship off the sandbank; but if 
she had been properly steered, she could have kept off it 
altogether. It now becomes clear why at the very outset, in 
shrewd prevision of the imminent danger, a certain manifold 
was introduced as the theatre of operation, although there w r as 
no reason for this in the pure domain-calculus. The subse- 
quent restriction of this field for our logical activities is by no 
means elegant. Whereas elsewhere logic may claim to have 
laws of unrestricted validity, we are here required to begin by 
delimiting a manifold with careful tests, and it is only then 
that we can move around inside it. 



D.THE RAMIFIED THEORY OF TYPES 

We come now to Russell's ramified theory of types in its first 
formulation (1908). 

48.21 A type is defined as the range of significance of a 
propositional function, i.e. as the collection of arguments for 
which the said function has values. Whenever an apparent 
variable occurs in a proposition, the range of values of the 
apparent variable is a type, the type being fixed by the 
function of which "all values" are concerned. The division of 
objects is necessitated by the reflexive fallacies which other- 
wise arise. These fallacies, as we saw, are to be avoided by 
what may be called the "vicious circle principle"; i.e. "no 
totality can contain members defined in terms of itself". This 
principle in our technical language becomes: "whatever con- 

393 



MATHEMATICAL VARIETY OF LOGIC 

tains an apparent variable must not be a possible value of 
that variable". Thus whatever contains an apparent variable 
must be of a different type from the possible values of that 
variable; we will say that it is of a higher type. Thus the 
apparent variables contained in an expression are what 
determines its type. This is the guiding principle in what 
follows. 

Propositions which contain apparent variables are generated 
from such as do not contain these apparent variables by 
processes of which one is always the process of generalization, 
i.e. the substitution of a variable for one of the terms of the 
proposition, and the assertion of the resulting function for all 
possible values of the variable. Hence a proposition is called 
a generalized proposition when it contains an apparent variable. 
A proposition containing no apparent variable we will call an 
elementary proposition. It is plain that a proposition con- 
taining an apparent variable presupposes others from which 
it can be obtained by generalization; hence all generalized 
propositions presuppose elementary propositions. In an elemen- 
tary proposition we can distinguish one or more terms from 
one or more concepts; the terms are whatever can be regarded 
as the subject of the proposition, while the concepts are 
the predicates or relations asserted of these terms. The terms 
of elementary propositions we will call individuals; they form 
the first and lowest type. . . . 

By applying the process of generalization to individuals 
occuring in elementary propositions, we obtain new proposi- 
tions. The legitimacy of this process requires only that no 
individuals should be propositions. That this is so, is to be 
secured by the meaning we give to the word individual. We 
may define an individual as something destitute of complexity ; 
it is then obviously not a proposition, since propositions are 
essentially complex. Hence in applying the process of generali- 
zation to individuals, we run no risk of incurring reflexive 
fallacies. 

Elementary propositions together with such as contain 
only individuals as apparent variables we will call first-order 
propositions. These form the second logical type. 

We have thus a new totality, that of first-order propositions. 
We can thus form new propositions in which first-order 
propositions occur as apparent variables. These we will call 
second-order propositions; these form the third logical type. 

394 



ANTINOMIES AND TYPES 

Thus e.g. if Epimenides asserts "all first-order propositions 
affirmed by me are false", he asserts a second-order proposition; 
he may assert this truly, without asserting truly any first- 
order proposition, and thus no contradiction arises. 

The above process can be continued indefinitely. The 
n + 1th logical type will consist of all propositions of order n, 
which will be such as contain propositions of order n - 1 but 
of no higher order, as apparent variables. The types obtained 
are mutually exclusive, and thus no reflexive fallacies are 
possible so long as we remember that an apparent variable 
must always be confined within one type. 

In practice, a hierarchy of functions is more convenient 
than one of propositions. Functions of various orders may be 
obtained from propositions of various orders by the method of 
substitution. If p is a proposition, and a a constituent of p, let 
'p/a;#' denote the proposition which results from substituting 
x for a wherever a occurs in p. Then p/a, which we will call a 
matrix, may take the place of a function; its value for the 
argument x is p/a;x, and its value for the argument a is p. . . . 
The order of a matrix will be defined as being the order of the 
proposition in which the substitution is effected. 

In the same connection we adduce a further text, from the Prin- 
cipia, which has two points of interest. First, the idea expounded is 
very close to one expounded by Paul of Venice (35.41—43; 35.49 ft".); 
second, the text contains the essentials of the simple theory of types. 

48.22 An analysis of the paradoxes to be avoided shows 
that they result from a certain kind of vicious circle. The 
vicious circles in question arise from supposing that a collec- 
tion of objects may contain members which can only be 
defined by means of the collection as a whole. Thus, for exam- 
ple, the collection of propositions will be supposed to contain a 
proposition stating that "all propositions are either true or 
false". It would seem, however, that such a statement could 
not be legitimate unless "all propositions" referred to some 
already definite collection, which it cannot do if new proposi- 
tions are created by statements about "all propositions". We 
shall, therefore, have to say that statements about "all proposi- 
tions" are meaningless. More generally, given any set of objects 
such that, if we suppose the set to have a total, it will contain 
members which presuppose this total, then such a set cannot 
have a total. By saying that a set has "no total", we mean, 

395 



MATHEMATICAL VARIETY OF LOGIC 

primarily, that no significant statement can be made about 
"all its members". 



E. SYSTEMATIC AMBIGUITY 

The application of the theory of types to expressions containing 
the word 'true' and suchlike issues at once in the thesis that words 
of this kind are ambiguous. This is formulated in the Principia as 
follows : 

48.23 Since "(x),yx" involves the function <px, it must 
according to our principle, be impossible as an argument to 
9. That is to say, the symbol '9 { (x) . 9^}' must be meaning- 
less. This principle would seem, at first sight, to have certain 
exceptions. Take, for example, the function "p is false", and 
consider the proposition "(p) . p is false". This should be a 
proposition asserting all propositions of the form "p is false". 
Such a proposition, we should be inclined to say, must be 
false, because "p is false" is not always true. Hence we should 
be led to the proposition 

"{(p) . p is false } is false", 
i.e. we should be led to a proposition in which "(p) . pis false" 
is the argument to the function 'p is false' which we had 
declared to be impossible. Now it will be seen that "(p) . p 
is false", in the above, purports to be a proposition about all 
propositions, and that, by the general form of the vicious- 
circle principle, there must be no propositions about all 
propositions. Nevertheless, it seems plain that, given any 
function, there is a proposition (true or false) asserting all its 
values. Hence we are led to the conclusion that "p is false" 
and ll q is false" must not always be the values, with the argu- 
ments p and q, for a single function "p is false". This, however 
is only possible if the word "false" really has many different 
meanings, appropriate to propositions of different kinds. 

That the words "true" and "false" have many different 
meanings, according to the kind of proposition to which they 
are applied, is not difficult to see. Let us take any function 
yx, and let a be one of its values. Let us call the sort of truth 
which is applicable to (pa " first truth". (This is not to assume 
that this would be first truth in another context; it is merely 
to indicate that it is the first sort of truth in our context.) 
Consider now the proposition (x) . (px. If this has truth of the 

396 



ANTINOMIES AND TYPES 

sort appropriate to it, that will mean that every value yx 
has "first truth". Thus if we call the sort of truth that is appro- 
priate to (x) . (px "second truth", we may define "{(#) . 9.x} has 
second truth" as meaning "every value for <px has first truth", 
i.e. "(x) . (9 x has first truth)". 

48.24 It will be seen that, according to the above hierarchy 
no statement can be made significantly about "all a-functions" 
where a is some given object. Thus such a notion as "all 
properties of a", meaning "all functions which are true with 
the argument a", will be illegitimate. We shall have to dis- 
tinguish the order of function concerned. We can speak of 
"all predicative properties of a", "all second-order properties 
of a", and so on. (If a is not an individual, but an object of 
order n, "second-order properties of a" will mean "functions 
of order n + 2 satisfied by a".) But we cannot speak of "all 
properties of a". In some cases, we can see that some statement 
will hold of "all nth-order properties of a", whatever value n 
may have. In such cases, no practical harm results from 
regarding the statement as being about "all properties of a", 
provided we remember that it is really a number of statements, 
and not a single statement which could be regarded as assign- 
ing another property to a, over and above all properties. 
Such cases will always involve some systematic ambiguity, 
such as that involved in the meaning of the word 'truth', as 
explained above (cf. 48.23). Owing to this systematic ambi- 
guity, it will be possible, sometimes, to combine into a single 
verbal statement what are really a number of different 
statements, corresponding to different orders in the hierarchy. 
This is illustrated in the case of the liar, where the statement 
"ally's statements are false" should be broken up into different 
statements referring to his statement of various orders, and 
attributing to each the appropriate kind of falsehood. 

It should be clear that the authors of the Principia do not go far 
enough in suggesting that these different statements are combined 
only into a single verbal statement; for all the statements in question 
evidently share the same formal structure. We have in fact a case 
of isomorphy (47.41). It is remarkable that the name used for this 
kind of isomorphy, 'systematic ambiguity', is an exact translation 
of the common Scholastic expression aequivocatio a consilio. synony- 
mous with 'analogy' (28.18 ff.); for isomorphy is precisely analogy. 



397 



MATHEMATICAL VARIETY OF LOGIC 

F.THE AXIOM OF REDUCIBILITY 

Closely connected with the ramified theory of types and systema- 
tic ambiguity is the axiom of reducibility used in the Principia. 
The expression 'predicative function' used in this context is defined 
thus: 

48.25 We will define a function of one variable as predi- 
cative when it is of the next order above that of its argument, 
i.e. of the lowest order compatible with its having that argu- 
ment. If a function has several arguments, and the highest 
order of function occurring among the arguments is the nth, 
we call the function predicative if it is of the n + 1th order, 
i.e. again, if it is of the lowest order compatible with its 
having the arguments it has. A function of several arguments is 
predicative if there is one of its arguments such that, when the 
other arguments have values assigned to them, we obtain a 
predicative function of the one undetermined argument. 

48.24 follows soon after, and the passage continues: 

48.26 The axiom of reducibility is introduced in order to 
legitimate a great mass of reasoning, in which, prima facie, we 
are concerned with such notions as 'all properties of a' or 
'all a-functions', and in which, nevertheless, it seems scarcely 
possible to suspect any substantial error. In order to state 
the axiom, we must first define what is meant by 'formal 
equivalence'. Two functions <px, tyx are said to be 'formally 
equivalent' when, with every possible argument x, yx is 
equivalent to tyx, i.e. <p# and <\>x are either both true or both 
false. Thus two functions are formally equivalent when they 
are satisfied by the same set of arguments. The axiom of 
reducibility is the assumption that, given any function <px, 
there is a formally equivalent predicative function, i.e. there is 
a predicative function which is true when yx is true and 
false when yx is false. In symbols, the axiom is: 

h : (3 <J/) : <p x .=jb . tyl x . 
For two variables, we require a similar axiom, namely: 
Given any function <p (x, y), there is a formally equivalent 
predicative function, i.e. 

h : (3+) : y(x,y) . =*,y . t|i ! (x, y). 

This text has two points ol interest. First, it contains quantified 
predicates (functors), like Peirce's definition of identity (44.26); 
this must eventually lead to the construction of the so-called higher 
predicate calculus, though this is not to be found till some time after 

398 



ANTINOMIES AND TYPES 

the Principia. In the Principia, with few exceptions, only argu- 
ments, not functors, are quantified. Secondly, it is clear that the 
axiom of reducibility is a further case of isomorphy, so that one 
usually speaks, with systematic ambiguity, of the reducibility axiom. 
The authors of the Principia themselves admit (14.27) that it is not 
at all evident, and as an axiom it is not proved. Its justification lies 
wholly in its convenience, which is no great logical recommendation. 
Hence logicians have concerned themselves to dispense with it. 

G.SIMPLE THEORY OF TYPES 
1. Chwistek 

L. Chwistek was the first (1921) to propose discarding this axiom 
and the complications of the ramified theory of types: 

48.28 Thus a theory of logical types seems to be an absolu- 
tely necessary basis of every modern formal logic which is 
going to retain the fundamental operations of the algebra of 
logic. . . . 

But when we ask whether Russell's theory of types fully 
serves its purpose, we must say this. According to this theory 
every object has a determinate logical type and every domain 
of validity of an argument consists of objects which belong to 
the same logical type. 

But the converse does not hold, since an object of given type 
can belong to the domain of validity of arguments of functions 
with different types. Two functions which have different 
types and the same domain of validity of their arguments we 
shall call, with Russell, functions of different ranks (i.e. orders; 
cf. 48.21). 

Russell now posits that there is a lowest rank of functions, 
those characterized by containing no apparent variables, i.e. 
which can be envisaged without use of the concepts 'for all 
x' and 'for some x\ . . . 

And for Russell's theory of types a certain axiom (the so- 
called axiom of reducibility) is essential. . . . 

I should now like to show that the adoption of this axiom 
straightway leads to the reconstruction of Richard's antinomy. 

There follows the reconstruction of this antinomy within a system 
containing the axiom of reducibility, and Chwistek goes on : 

48.29 Thus the axiom of reducibility appears as a contra- 
dictory supposition, and hence we cannot agree without 
reserve to Russell's theory of types. 

399 



MATHEMATICAL VARIETY OF LOGIC 

If we were to share the hesitations of many logicians about 
the theoretical worth of Richard's method, it would be clear 
that there is no real basis for distinguishing between ranks of 
propositional functions. In any case Russell's theory of types 
needs a critical overhaul. . . . 

48.30 Finally I may be allowed to say something about 
the results of my endeavours over many years to reconstruct 
the system of Russell and Whitehead without the axiom of 
reducibility. 

If we deny any theoretical value to Richard's antinomy, 
formal logic, obtains an essential simplification, since, though 
we still have to do with types, we no longer have orders of 
functions, so that we can briefly speak of 'all properties' of 
a thing. In this way we can at once obtain a system, different 
from that of Russell and Whitehead, having the same theo- 
retical value. 

But the freedom from contradiction of this systems remains 
very doubtful, and apart from this, the following most 
important question remains to be answered : 

Is a system of formal logic possible, which would be based on 
the general theory of types, without the axiom of reducibility, and 
which would not have to adopt any new axioms? 

I am in a position to give an affirmative answer to this 
question, having succeeded in constructing such a system, and 
in particular in establishing the theory of cardinal and induc- 
tive numbers without any additional hypothesis beyond what 
is already present in the system of Russell and Whitehead. 

2. Ramsey 

Ramsey reached the same conclusion in 1925. He lays particular 
emphasis on the distinction between logical and semantic antino- 
mies: 

48.31 It is not sufficiently remarked, and the fact is entirely 
neglected in Principia Mathematica, that these contradictions 
fall into two fundamentally distinct groups, which we will 
call A and B. The best known ones are divided as follows: - 

A. (1) The class of all classes which are not members of 
themselves. 

(2) The relation between two relations when one does 
not have itself to the other. 

(3) Burali Forti's contradiction of the greatest ordinal. 

B. (4) 'I am lying'. 

400 



ANTINOMIES AND TYPES 

(5) The least integer not nameable in fewer than 
nineteen syllables. 

(6) The least indefinable ordinal. 

(7) Richard's Contradiction. 

(8) Weyl's Contradiction about 'heterologisch' *. 
Footnote of Ramsey's: *For the first seven of these see 

Principia Mathematical, 1 (1910), 63. For the eighth see Weyl, 
Das Kontinuum, 2. 

The principle according to which I have divided them is of 
fundamental importance. Group A consists of contradictions, 
which, were no provision made against them, would occur 
in a logical or mathematical system itself. They involve only 
logical or mathematical terms such as class and number, and 
show that there must be something wrong with our logic or 
mathematics. But the contradictions of Group B are not 
purely logical, and cannot be stated in logical terms alone, for 
they all contain some reference to thought, language, or 
symbolism, which are not formal but empirical terms. So they 
may be due not to faulty logic or mathematics, but to faulty 
ideas concerning thought and language. If so, they would not 
be relevant to mathematics or to logic, if by 'logic' we mean 
a symbolic system, though, of course, they would be relevant 
to logic in the sense of the analysis of thought. 

48.32 A theory of types must enable us to avoid the con- 
tradictions; Whitehead and Russell's theory consisted of 
two distinct parts, united only by being both deduced from 
the rather vague 'Vicious-Circle Principle'. The first part 
distinguished propositional functions according to their 
arguments, i.e. classes according to their members; the second 
part created the need for the Axiom of Reducibility by 
requiring further distinctions between orders of functions with 
the same type of arguments. 

We can easily divide the contradictions according to which 
part of the theory is required for their solution, and when we 
have done this we find that these two sets of contradictions 
are distinguished in another way also. The ones solved by the 
first part of the theory are all purely logical; they involve 
no ideas but those of class, relation and number, could be 
stated in logical symbolism, and occur in the actual develop- 
ment of mathematics when it is pursued in the right direction. 

* more exactly, Grelling's. 

401 



MATHEMATICAL VARIETY OF LOGIC 

Such are the contradiction of the greatest ordinal, and that of 
the class of classes which are not members of themselves. 
With regard to these Mr. Russell's solution seems inevitable. 
On the other hand, the second set of contradictions are 
none of them purely logical or mathematical, but all involve 
some psychological term, such as meaning, defining, naming 
or asserting. They occur not in mathematics, but in thinking 
about mathematics; so that it is possible that they arise not 
from faulty logic or mathematics, but from ambiguity in the 
psychological or epistemological notions of meaning and 
asserting. Indeed, it seems that this must be the case, because 
examination soon convinces one that the psychological term 
is in every case essential to the contradiction, which could not 
be constructed without introducing the relation of words to 
their meaning or some equivalent. 

It can be seen how sharply Ramsey distinguishes between object- 
and metalanguage. But it is to be noted that he does not use these 
expressions, and can only conceive the domain of relations between 
signs and significates psychologically or epistemologically. In the 
course of later developments, this domain became separated (by 
Lesniewski, and above all by Tarski) and denoted as that of seman- 
tics. 

The simple theory of types proposed in the last text is not at all 
such a simple matter as an uninitiated reader might at first suppose. 
The axiom of reducibility is replaced by so-called pseudo-definitions, 
or alternatively, by axioms about the existence of classes, and the 
freedom from contradiction of these axioms is not an easy question; 
so that the theory under consideration does not lack its problems. * 

We cannot here pursue this question, nor the later development of 
the whole matter. 



§ 49. SOME RECENT DOCTRINES 

As an appendix, we shall speak briefly in this paragraph of a few 
of the problems and doctrines that have arisen since the Principia. 
It will only be a matter of some examples of the enormous devel- 
opment characteristic of the fourth period of mathematical logic 
(vide § 37), which does not properly fall within our scope. 

We must forgo speaking of doctrines which constitute a break 
with tradition, such as those concerning the 'natural logics' 

*Thanks are due to Prof. E. W. Beth for important information on this point. 

402 



REGENT DOCTRINES 

as also of the numerous and often pioneer Insights into semantic 
questions. None of this yet belongs to history, and has, moreover, 

only slight connection with what has so far been expounded. 

Up to 1918 all mathematical logicians - unlike the Megarians, 
Stoics and Scholastics - used only one notion of implication, the 
Philonian (20.07) or material (41.13f). Hence the mathematical 

logic of that time was exclusively assertoric, a logic without modali- 
ties, or in other words, a two-valued logic. It supposed only two 
values, truth and falsity. The only exception, to our knowledge, is 
the system of McColl (49.01). In 1918 G. I. Lewis introduced a new 
notion of implication and with it a modal logic, since when a whole 
series of non-Philonian implications have been propounded and 
elaborated, i.e. many-valued logics have been developed. We mention 
here two such systems, Lewis's system of 'strict implication' (1918) 
and Lukasiewicz's three-valued logic (which he discovered in 1917). 
Godel's famous theorem, his account of which we give in con- 
clusion, belongs to another domain of problems. 



A. STRICT IMPLICATION: LEWIS 

Lewis formulated the idea of 'strict implication' as early as 1913 
(49.02). The following texts date from 1918: 

49.03 The fundamental ideas of the system are similar to 
those of MacColl's Symbolic Logic and its Applications. They 
are as follows : 

1. Propositions: p, q, r, etc. 

2. Negation: - p, meaning 'p is false'. 

3. Impossibility: ~ p, meaning 'p is impossible', or 'It is 
impossible that p be true'. 

4. The logical product: p x q or pq, meaning 'p and q both', 
or 'p is true and q is true'. 

5. Equivalence: p = q, the defining relation. 

Systems previously developed, except MacColl's, have only 
two truth-values, 'true' and 'false'. The addition of the idea 
of impossibility gives us five truth-values, all of which are 
familiar logical ideas : 

(1) p, 'p is true'. 

(2) - p, 'p is false'. 

(3) ^ p, 'p is impossible'. 

(4) - ^ p, 'It is false that p is impossible' i.e., 'p is pos- 
sible'. 

(5) ^ - p, 'It is impossible that p be false' i.e., 'p is 
necessarily true'. 

403 



MATHEMATICAL VARIETY OF LOGIC 

49.04 The dyadic relations of propositions can be defined 
in terms of these truth-values and the logical product, pq. 

1.01 Consistency, p o q = - ~ [pq). Def. 

ro (pq), 'It is impossible that p and q both be true' would be 
l p and q are inconsistent'. Hence - ~ (pq), 'It is possible 
that p and q both be true', represents 'p and q are consistent'. 

1.02 Strict Implication. p -$ q = ~ (p -q). Def. 

1.03 Material Implication. p c q = - (p -q). Def. 

1.04 Strict Logical Sum. p a q = ~ (-p -q). Def. 

1.05 Material Logical Sum. p + q = - (-p -q). Def. 

1.06 Strict Equivalence, (p = q) = (p -} q) (q -3 p). Def. 



1.07 Material Equivalence, (p = q) = (p d g) (g Dp). Def. 

Lewis thought that his 'strict' implication resembled the ordinary 
'if - then' more closely than does material implication. But this 
seems not to be the case. While avoiding the classical 'paradoxes of 
implication' (31.411-412. 43.22 [01] and [36]), it brings its own 
paradoxes (cf. 31.141), as the following theorems of Lewis's system 
show: 

49.05 . . . an . . . analogy holds between material implication, 
p d q, and strict implication, p -3 q. 

3.41 (p cq) S (-q c -p) 2.62 (p 4 q) -i (-</ -! -p) 

If p materially implies g, If p strictly implies q, then 

then 'g is false' materially 'q is false' strictly implies 

implies 'p is false'. 'p is false'. 

3.42 -p ^(p d g) 3.52 ~ p ^ (p ^ q) 

If p is false, then p mate- If p is impossible (not 

rially implies any proposition self-consistent, absurd), then 

q. p strictly implies any propo- 

3.43 ( pc -p) -^ -p sition, g. 

If p materially implies its 3.53 (p -3 -p) -3 ~ p 
own negation, then p is false. If p strictly implies its own 

3.44 [p c (q c r)] -3 [5 c (p c r)] negation, then p is impos- 
sible (not self-consistent, ab- 
surd). 

3.54 [pl(qcr)]4[q4(pcr)] 

We may add certain further theorems which are consequen- 
ces of the above. 

3.45 p -8 (q cp) 3.55 ~ -p -3 (q ■* p) 

If p is true, then every If p is necessarily true, 

404 



RECENT DOCTRINES 

proposition, g, materially then p is strictly implied by 

implies p. any proposition, q. 

More such laws follow in the original. 

B. MANY-VALUED LOGICS: LUKASIEWICZ 

Many-valued logies were an important discovery. J. Lukasiewicz 
discovered a system of this kind in 1917 and lectured on it to the Philo- 
sophical Society in Lwow in 1920 (49.06). In the same year E. L. 
Post independently published another such system (49.07). We cite 
a passage on the subject from a lecture by Lukasiewicz in 1929, 
for its comparatively ready comprehensibility. 

49.08 One could, however, adopt a position which is incom- 
patible with the principle of two-valued logic. According to 
this position logical propositions can have values other than 
truth and falsity. A proposition of which we do not know 
whether it is true or false, may have no determinate value of 
truth or falsity but may have a third, undetermined value. 
One may think, for instance, that the proposition 'I shall be 
in Warsaw in a year's time' is neither true nor false, but has 
the third, undetermined value, which we can symbolize by' %'. 
But we can go still further and ascribe infinitely many values 
to propositions, values which lie between falsity and truth. In 
this case we should have an analogy with the calculus of 
probability, in which we ascribe infinitely many degrees of 
probability to different events. In this way we should obtain 
a whole heap of many-valued logics ; three-valued, four-valued, 
etc. and finally an infinitely-valued logic. Symbols other than 
T and '0', such as are used in proofs of independence, would 
thus correspond with propositions having different degrees of 
truth, in logics with a corresponding number of values. Actu- 
ally, it is the method of proving the independence of proposi- 
tions in the theory of deduction which has occasioned our 
researches into many-valued logics. 

In three-valued logic tables for implication and negation 
must be set up, analogous to those which we have in two- 
valued logic. Those given here seem to me very intuitive. 

Thus every meaningful expression 
would be true in threevalued logic if 
for all substitutions of the symbols 
*0\ i y 2 \ T for variable expressions. 
they yielded always '1' when the 

405 



c 





% 


i 


AT 





1 


i 


i 


1 


y 2 


y 2 


i 


i 


y 2 


i 





y 2 


i 






MATHEMATICAL VARIETY OF LOGIC 

reduction had been completed according to the accompanying 
table. It is easy to verify that in a three-valued logic thus 
understood, our axioms 1 (' CCpqCCqrCpr') and 3 {'CpCNpq') 
are true. But axiom 2 ('CCNppp') is not true. For the sub- 
stitution p/% gives us 

ccn y 2 y 2 y 2 = cc y 2 y 2 y 2 = c l y 2 = y 2 . 

It follows from the table given, that every sentence of three- 
valued logic is also a sentence of two-valued logic (but 
evidently not conversely). Three-valued logic can be exhibited 
as an axiomatic system, in the same way as that in which 
we have thought of the (two-valued) theory of deduction. 
Then we should not need to appeal to the given table proving 
sentences. 

In infinitely-valued logic we suppose that propositions can 
take infinitely many values; we co-ordinate them with the 
rational numbers satisfying the condition < x < 1. For 
such a logic it is evidently impossible to set up a table, since 
this would have to have infinitely many rows and columns. 
We determine the properties of implication and negation for 
infinitely-valued logic in the following way, premising that p 
and q are rational numbers in the interval - 1 : 

if p ^ q, then Cpq = 1 ; if p > q, then 
Cpq = 1 - p + q; Np = 1 - p. 
From these equations there follow the properties of implica- 
tion and negation which they were given in three-valued logic. 
From the said equations one can see that when the arguments 
of implication and negation do not go beyond the interval 
of the rational numbers (0, 1) *, then the values of implication 
and negation, too, do not go beyond this interval. 

In the many-valued logic we adopt the following defini- 
tions : 

Apq = CCpqq, Kpq = NANpNq, Epq = KCpqCqp. 
We have already spoken of the definitions given for alterna- 
tion and equivalence. The definition of conjunction is based 
on the De Morgan laws. 

Infinitely-valued logic is a proper part of two-valued 
logic; mostly, those sentences of two valued logic are not 
true in it, on which certain kinds of apagogic inferences are 
based. 

The relationship of many-valued logics to the two-valued 

'Lukasiewicz has the parentheses. 

406 



RECENT DOCTRINES 

reminds one of the relationship of non-Euclidean geometries 
to the geometry of Euclid. Like the non-Euclidean geome- 
tries, so too the many-valued logics are internally consistent, 
but different from the two-valued logic. Without deciding 
the question of the truth of one of these logics, we remark 
that the two-valued logic has this superiority: it is much 
simpler than the many-valued logics. In any case, the many- 
valued logics have proved useful, in leading to the method 
of finding independence. 

Wajsberg axiomatized three-valued logic in 1931 (49.09;. The 
question of the interpretation of these systems is still not clarified. 
While many logicians - e.g. Bernays (49.10) - think that they 
admit of no ready interpretation and so can hardly count as 'logics', 
H. Reichenbach has shown that the theory of Quantum Mechanics 
can be axiomatized on the basis of Lukasiewicz's three-valued logic, 
which cannot be done on the basis of two-valued logic (49.11). 

C.GODEL'S THEOREM 

As a final text from the history of problems treated in mathemati- 
cal logic we choose K. Godel's celebrated paper of 1931. It belongs to 
methodology rather than logic, but its great importance for the 
latter justifies its finding a place here. 

49.12 As is well known, the development of mathematics in 
the direction of greater exactness has led to the formalization 
of many of its domains in such a way that proof can be carried 
through in accordance with a few mechanical rules. The most 
extensive formal systems so far constructed are the system 
of Principia Mathematica [PM] 1 on the one hand, and the 
axiom-system for set-theory due to Zermelo and Fraenkel 
[further developed by J. v. Neumann] on the other. Both 
these systems are so comprehensive that all contemporary 
methods of proof used in mathematics have been formalized 
in them, i.e. have been reduced to a few axioms and rules of 
inference. Hence the conjecture suggests itself that these 
axioms and rules of inference are sufficient to decide all 
mathematical questions which can be completely formally 
expressed in the said systems. In what follows, it will be 
shown that this is not the case, but that in both the systems 
mentioned there are even relatively simple problems from the 
theory of the familiar integers 4 which cannot be decided on 
the basis of the axioms. This circumstance is not due at all 
to the special nature of these systems, but holds for a very 

407 



MATHEMATICAL VARIETY OF LOGIC 

large class of formal systems, to which there belong in 
particular all those formed from these two by the addition of 
a finite number of axioms 5 , it being presumed that no false 
theorems of the kind specified in foot-note 4 become provable 
through the added axioms. 

We first sketch, before going into details, the main ideas 
of the proof, naturally without laying claim to exactness. 
The formulae of a formal system [we here confine ourselves to 
the system of PM] are finite material sequences of basic 
symbols [variables, logical constants, and parentheses or 
punctuation-marks], and it is easy to specify exactly which 
sequences of basic symbols are meaningful formulae and 
which are not. 6 Analogously, from the formal point of view 
proofs are only finite sequences of formulae [with definite 
statable properties]. For metamathematical purposes it is 
naturally indifferent what objects are taken as basic sym- 
bols, and we opt for using natural numbers 7 as such. Accord- 
ingly, a formula becomes a finite sequence of natural num- 
bers 8 and a proof becomes a finite sequence of finite sequences 
of natural numbers. Metamathematical concepts [theorems] 
thus become concepts [theorems] about natural numbers or 
sequences of such, 9 and consequently expressible [at least in 
part] in the symbols of the system PM itself. In particular, 
it can be shown that the concepts 'formula', 'proof, 'provable 
formula', are definable in the system PM, i.e. it is possible 
to produce a formula F(v), for instance, in PM, having a 
free variable v [of the type of a sequence of numbers], 10 such 
that F(v) states when meaningfully interpreted : v is a provable 
formula. Now we state an undecidable sentence of the system 
PM, i.e. a sentence A such that neither A nor not- A is provable, 
in the following way: 

A formula of PM with just one free variable, and that of the 
type of the natural numbers [class of classes] we will call 
a class-symbol. We think of class-symbols as somehow ordered 
in a sequence 11 , denote the n-th by R{n), and note that the 
concept 'class-symbol', as also the ordering relation R, can 
be defined in PM. Let a be an arbitrary class-symbol; by 
[a; n] we denote the formula which arises from the class- 
symbol a when the free variable is replaced by the symbol 
for the natural number n. The ternary relation x — [y; z] 
can also be shown to be definable in PM. Now we define a 
class K of natural numbers, as follows : 

408 



RECENT DOCTRINES 

n zK ^Ttew'lB (n); n] n ' d [1] 

[where Bew x signifies: x is a provable formula]. Since the 
concepts occurring in the definiens are all definable in PM, 
so is the concept K, formed from them, i.e. there is a class- 
symbol S 12 such that the formula [S; n] states when meaning- 
fully interpreted, that the natural number n belongs to K. 
As a class-symbol, S is identical with a determinate fi(q), i.e. 

S = R{q) 
holds for a determinate natural number q. We now show that 
the sentence [B(q); q] 13 is undecidable in PM. For on the 
supposition that the sentence [B(q); q] was provable, it would 
also be true, i.e. according to what has been said q would 
belong to K, and so by [1] Bew [B(q); q] would hold, in con- 
tradiction to the supposition. But if on the contrary the 
negation of [B(q); q] was provable, then n z K would hold, i.e. 
Bew [B(q); q]. Thus [B(q); q] as well as its negation would 
be provable, which is again impossible. 

The analogy of this argument with the antinomy of Richard 
(48.14) leaps to the eye; it is also closely comparable with 
the 'Liar' 14 (23.10ff., 35.11 ff., 48.10), since the undecidable 
sentence [B(q); q] states that q belongs to K, i.e., by [1], that 
[B(q); q] is not provable. We thus have a sentence that states 
its own unprovability. 15 The method of proof just expounded 
can be applied to every formal system which, firstly, when 
meaningfully interpreted, disposes of sufficient means of 
expression to define the concepts occurring in the foregoing 
considerations [in particular the concept 'provable formula'], 
and in which, secondly, every provable formula is also 
meaningfully true. The exact carrying out of the foregoing 
proof that follows will have the task, among others, of replac- 
ing the second of those requirements by a purely formal and 
much weaker one. 

From the observation that [B(q); q] states its own unpro- 
vability, it at once follows that [B(q); q] is true, since [B(q); q] 
is indeed unprovable [since undecidable]. The sentence that 
is undecidable in the system PM has thus been decided by 
metamathematical considerations. The exact analysis of this 
remarkable circumstance leads to surprising results concern- 
ing proofs that formal systems are free from contradiction. 
These will be considered more closely in Section 4 Theorem 
(XI). 



409 



MATHEMATICAL VARIETY OF LOGIC 

1 Cf. the summary of the results of this work in Anzeiger der 
Akad. d. Wiss. in Wien [math.-naturw. KL] 1930 Nr. 19. . . . 

4 I.e. more exactly, there are undecidable sentences in 
which besides the logical constants - [not], v [or], (x) [for 
all], = [identical with], there occur no other concepts than + 
[addition], . [multiplication], both applying to natural 
numbers, the prefix (x) also being applicable only to natural 
numbers. 

5 Only such axioms being reckoned distinct in PM as are 
not differentiated merely by alteration of types. 

6 Here and in what follows we always understand by 
'formula of PM' a formula written without abbreviations 
[i.e. without application of definitions]. Definitions serve 
only to procure shorter expressions and are therefore dispen- 
sable with in principle. 

7 I.e. we model the basic symbols on the natural numbers 
in an unambiguous way. . . . 

8 I.e. a co-ordination of a segment of the number series 
with natural numbers. [Numbers cannot be put in a spatial 
order.] 

9 In other words: the procedure described yields an iso- 
morphic model of the system PM in the domain of arithmetic, 
and all metamathematical considerations can be equally 
well treated in this isomorphic model. This is done in the 
sketched proof that follows, i.e. by 'formula', 'sentence', 
'variable' etc. we are always to understand the corresponding 
objects in the isomorphic model. 

10 It would be quite easy [only rather lengthy] actually to 
write out this formula. 

11 E.g. according to increasing sums of terms, and for 
equal sums, lexicographically. 

lla The stroke across the top denotes negation. 

12 Again there is not the least difficulty in actually writing 
out the formula S. 

13 Note that l [R{q); q)]' or '[S;q]\ which means the same 
thing, is merely a metamathematical description of the unde- 
cidable sentence. But as soon as the formula S has been 
ascertained, it is of course possible to determine also the 
number q, and so effectively to write out the undecidable 
sentence itself. 

14 Every epistemological antinomy can be subjected to an 
undecidability-proof of this kind. 

410 



RECENT DOCTRINES 

15 Contrary to intuition, such a sentence involves nothing 
essentially circular, since it first states the unprovability of 
a quite definite formula [viz. the q-th in the lexicographical 
ordering on a definite substitution], and only then [as it 
were casually] does it appear that this formula is precisely 
the one in which it has itself been expressed. 

D. COMBINATORY LOGIC 

In a lecture at Gottingen in 1920 (edited by H. Behmann in 1924 
(49.13)) M. Schonfinkel laid the foundations of a new development 
which in Gertain respects has no parallel in other varieties of logic. We 
have seen successful efforts made to reduce the number of undefined 
propositional functors (§ 43 E) and to define the existential cjuantifier 
in terms of the universal (cf. 44.14) ; the new endeavour is to dispense 
with the use of variables, part of the primitive capital of formal 
logic (cf. § 9 B, a, (aa)). Schonfinkel states his program as follows: 

44.14 It is agreeable to the axiomatic method . . . that one 
should not only do one's best to restrict the number and 
content of the axioms, but also to reduce as far as possible the 
number of the basic undefined concepts. . . . The progress 
so far made along this road prompts the effort towards a 
further step ... to try to eliminate even the remaining basic 
concepts of proposition, propositional function and variable 
by suitable reduction. To investigate and follow up such a 
possibility more closely would be valuable not only from the 
point of view of methodological endeavour to obtain the 
greatest possible unity of thought, but also from a certain 
philosophical or, if preferred, aesthetic point of view. For 
the variable in a logical proposition serves only as a mark 
distinctive of certain argument-places and operators as 
mutually relevant, and hence is to be characterized as a 
subsidiary concept, properly speaking unsuited to the purely 
constant, 'eternal' nature of a logical proposition. 

It seems to me very remarkable that even this goal we 
have set can be attained, in the sense, moreover, that the 
reduction is achieved by means of three basic concepts. 

Schonfinkel then introduces five 'functions' of general applicabi- 
lity, by the following definitional equations: 
the identity function I : Ix = x; 
the constant tunction C: Cxy —■ x; 
the permuting function T: T <&xy =<&yx; 
the compounding function Z : ZQjx = ®(x#); 

411 



MATHEMATICAL VARIETY OF LOGIC 

the amalgamating function S: S^yx = (Oaj) (-/x), 

and shows that S and C can be used to define the others : 

/ = sec, 

Z = S(CS)C, 

T = S(ZZS)(CC). 
The more properly logical incompatibility function U is then ex- 
plained by means of the Sheffer-stroke and the universal quantifier: 

Ufg = fx | x gx 

44.15 We now have the remarkable fact that every logical 
formula can be expressed not only by means of the several 
functions /, C, T, Z, S } U, but actually by means of C, S 
and U. 

As an example, he reduces the proposition: 

(f)(Eg)(x)fx&ir 

(symbolism of Hilbert, cf. § 41 H) to the form: 

U[S(ZUU)U] [S(ZUU) U]. 
Further researches in this direction have been made by H. B. Gurry 
(1930), J. B. Rosser (1935), A. Church (1941), R. Feys (1946) and 
P. Rosenbloom (1950). 

SUMMARY 

In summary, we can make the following points about the results 
achieved in the period of mathematical logic of which we have 
treated (up to the Principia) : 

1. Mathematical logic again presents us with a highly original 
variety of logic ; for in contrast to all the other known forms of this 
science, it proceeds constructively, i.e. by investigating logical laws 
in an artificial language that it has devised. Such artificial languages 
exhibit very simple syntactical and semantic relations, as compared 
with natural languages, with the result that formal logic has undergone 
a change very like that effected by Galileo in the domain of physics. 
Instead of the immediate, but complex facts, the simpler underlying 
connections can now be investigated. 

2. In comparison with that fundamental novelty, the constant, 
and after Boole deliberate, increase in formalism is less revolutionary, 
since formalism was highly developed also among the Stoics and 
Scholastics. But we know of no such thorough-going application of 
this method as in mathematical logic. 

3. With the help of the new principle of constructivity and of 
formalism, many old intuitions were recaptured and considerably 
developed in the course of this period, intuitions which had been lost 
to view in the barbaric 'classical' period. We may instance the 
concept of logical form, the distinction between language and 

412 



SUMMARY 

metalanguage, between propositions! and term-logic, the problem 
of semantic antinomies, and some aspects of other semantic problems. 

4. Further, we here find a long series of quite new discoveries. First 
and foremost, the problem of 'complete proof was posed and fully 
solved. The analysis of propositions was carried out by new means, 
though still in an Aristotelian sense, namely by applying the con- 
cepts of functor and argument, and by quantifiers. This led to prob- 
lems of many-place functors and plural quantification which had not 
been known before. The distinction between the logic of predicates and 
that of classes was not indeed quite new, having been treated in the 
Scholastic doctrine of supposition, but it now comes to be very 
sharply made. The logic of relations seems to be quite a new creation, 
notwithstanding some hints in Aristotle, Galen and the Scholastics, 
as also the theory of description and the logical antinomies. These are 
only a few examples. 

5. After all that, it must seem rather surprising that, up to and 
including the Principia, there is here less logical strictness (parti- 
cularly as concerns the distinction of language and metalanguage) 
than in the best texts of Megarian-Stoic and scholastic logic. Frege 
is the only exception. Even Lewis's explanations (cf. 49.05) lack 
precision. But this weakness was overcome after the Principia, so 
that logic attained once more a very high level of exactness. 

6. Finally, the large number of logical formulae stated and inves- 
tigated is characteristic of mathematical logic. Often this is due to a 
purely mechanical development and brings no interesting infor- 
mation, but often enough a much greater positive contribution is 
made than by the other forms of logic, especially in the logic of 
terms. 

Thus there can be no doubt that in this period formal logic once 
more attained one of its peaks of development. 



413 



PART VI 

The indian variety ojLogii 



§50. INTRODUCTION TO INDIAN LOGIC 



A. HISTORICAL SURVEY 

A sketch of the history of formal logic in India will be more 
intelligible to the reader if it is prefaced with some account of the 
basic evolution of Indian thought, which is but little known in the 
West. 

With some simplification we can put the beginning of systematic 
thought in India in the last centuries b.c. Various religious, psycho- 
logical and metaphysical conceptions are indeed known before that, 
but they first take on systematic form in the classical texts that 
survive from that time, texts called 'Sutra' by the Brahmins - the 
word means both a statement of doctrine and a work consisting of 
such. Of these texts six are Brahmanic in character, the Sdmkhya- 
karlka, the Yoga-, Purua-mimdmsd-, Veddnta-, Nydya- and Vaisesika- 
sulra. The last seems to have been first edited in about the first 
century a.d., the Nydya-siitra first about 200; their contents, 
however, are ascribed at least in part to an earlier period. Every one 
of these Sutras has occasioned a swarm of commentaries, commen- 
taries on commentaries, commentaries of the third order etc., and 
nearly the whole philosophical literature of India consists of commen- 
tarie? . The teachings of these schools can be characterized thus : 
Sdmkhya: dualistic ontology and cosmogony. 
Yoga: systematization of mythical and ascetical practice. 
Purva-mlmdmsd: monistic metaphysics. 
Nydya: epistemology, logic, and methodology. 
Vaisesika: realistic ontology and systematics. 

These bodies of teaching often supplement one another, e.g. those 
of the Vaisesika and Nydya. 

Besides the Brahmans there arose, among others, two further 
religious communions: the Buddhists and the Jins (for this spelling 
rather than 'Jains' see 50.01). Both took shape in the 6th century 
b.c. and in the centuries round the beginning of our era developed 
highly speculative systems of thought which first found expression 
in some fundamental texts. Buddhism is of over-riding importance 
to us. It is divisible into two great tendencies: the Hinayana (the 
little vehicle) and the Mahayana (the great vehicle). Within these 
two main streams again various schools arose. The chief schools of 
the Hinayana are the pluralistic-realistic Sarvastivada and the 
phenomenalistic Santrantika schools. In the Mahayana the first 
development was the negativistic relativism of the Madhyamikas. 
The movement culminated in the idealism of the Vijnanavada school. 
From among the followers of this last there should be mentioned at 
least the two brilliant brothers, the saintly Asariga, and Vasu- 

416 



SURVEY 

bandhu who was perhaps one of the most productive thinkers the 
history of philosophy has to show. 

Indian philosophy quickly developed permanent controversies, 
but also fruitful exchange of thought between the different schools. 
From the 8th century on, Buddhism lost ground and within Brah- 
manism the Vedanta gained the upper hand, mainly owing to a 
series of prominent thinkers of whom the most important is Sarikara, 
8-9th century. The final result, manifest even in the 10th century, is a 
unification: the Vedanta absorbed some doctrines of the other 
schools and also much Buddhist thought, and the controversies - 
such as that between the radical (advaila) pantheism of Sarikara 
and the moderate opinions of Ramanuja (11th century) took place 
entirely within the Vedantic school. 

Essentially, we can speak of three main periods of Indian logic 
which roughly coincide with the three millennia of its history: 

antiquity: approximately to the beginning of our era, the time 
of as yet unsystematic thought. 

classical period: the first millennium a.d., marked on the one 
hand by controversies between schools, on the other by the con- 
struction of developed systems. 

modern period: the second millennium a.d., with predominance 
of the Vedanta. 

B. EVOLUTION OF FORMAL LOGIC 

Formal logic (nydya-sdstra) developed in India, as in Greece, from 
the methodology of discussion. Such a methodology was already 
systematically constructed in the 2nd century b. c. The first ideas 
which can be said to be formal-logical occur indeed as early as the 
Vaisesika-sutra (1st century a.d.), but the history of Indian formal 
logic properly begins with the Nydya-sutra (edited in the 2nd century 
a.d.). This 'logical' sutra (so characterized by its very name) was 
the foundation of all Indian logical thought. 

After the final redaction of the Nydya-sutra the next five to six 
centuries display controversies between the Buddhist, Brahmanist 
and also Jinist logicians. In all three camps logic was keenly culti- 
vated. Among the most important thinkers are in the Naiyayikas*, 
Vatsyayana (5-6th cent.) (50.02),** Uddyotakara (7th cent.) "(50.03) 
and Vacaspati Misra (10th cent.)***; in the Vaisesikas primarily 

* I.e. among the followers of the Nyaya. The most important of other such 
names are 'Mimamsaka' for a follower of the Mimamsa, 'Vedantin' for a follower 
of the Vedanta, but simply 'Vaisesika' for a follower of the Vaisesika. 

** According to Shcherbatskoy, Vatsyayana might possibly be a contem- 
porary of Dignaga's, but D. Ingalls puts him in the 4th century (communica- 
tion by letter). 

*** This thinker has been generally ascribed to the 9th century, but I follow 
P. Hacker who puts him in the 10th (50.04). Prof. D. Ingalls was kind enough 
to draw my attention to Hacker's work. 

417 



INDIAN VARIETY OF LOGIC 

Prasastapada (5-6th cent.) (50.05), in the Mlmamsakas Kumarila 
(7th cent.) (50.06). Perhaps still more important than those is the 
Buddhist Vasubandhu (4^-5th cent.) (50.07) and his brilliant pupil 
Dignaga (5-6th cent.) (50.08). quite the greatest Indian logician, 
who founded an idealistic but unorthodox Vijnanavada-school. To 
this school there belong among others the commentator on Dignaga, 
Dharmakirti (7th cent.) (50.09), and his commentator Dharmottara 
(8-9th cent.) (50.10). In the same centuries occurred the crystalli- 
zation of formal logic which is plainly present in the 7th; a genuine 
and correct, though still in many ways elementary formal logic has 
developed from the methodology of public discussion. 

To the third period of Indian philosophy there corresponds a new 
epoch of logic, that of the Navya-Nyaya, the new Nyaya. Given 
shape by the Tattva-cintdmani, the great work of Gangesa (14th cent.) 
(50.11), this logic was developed with the utmost subtlety in a spirit 
remarkably like that of late western Scholasticism, though the basic 
ideas and methods are quite different. 

Of the innumerable logicians of this period the best known are 
Jayadeva (15th cent.) (50.12), Ragunatha (16th cent.) (50.13), 
Mathuranatha (50.14), Jagadisa (17th cent.) (50.15), and the author 
of a compendium not unlike the Summulae Logicales, Annambhatta 
(17th cent.) (50.16). 

Today the study of Indian logic has been re-introduced in India 
along with the resumption of speculative Vedantic thought (Sri 
Aurobindo). But it is not yet possible to form a judgment about 
this development. 

We set out the most important names and dates in a table : 

Pre-logical methodology of discussion 
THE OLD SCHOOL 
Nyaya-sutra (final redaction in 2nd century a.c.) 



naiyAyikas 
Vatsyayana (5-6 c.) 

Uddyotakara (7 c.) 

Vacaspari Misra (10 c. 
Udayana (end 10 c.) 
418 



BUDDHISTS 
Vasubandhu (4-5 c. 

Dignaga (5-6 c.) 



Dharmakirti (7 c.) 
Dharmottara (8-9 c.) 
Santaraksita (8 c.) 



OTHERS 



Prasastapada 

(5-6 c.) 
Kumarila (7 c.) 



(Prabhakara) 
gridhara (ca. 991) 



SURVEY 




THE NEW 


SCHOOL 


Garigesa 




14 c 


Jayadeva 




1425-1500 


Ragunatha 




ca. 1475-ca. 1550 


Mathuranatha 




ca. 1600- ca. 1675 


Jagadisa 




ca. 1600 


Annambhatta 




after 1600. 



C. STATE OF RESEARCH 

The present (1955) state of research in the field of Indian logic 
has a certain resemblance to that in the field of western Scholastic 
logic. Most of the logical texts are still unpublished, and many, 
especially the Buddhist ones, are only available in Tibetan or 
Chinese translations; many too are no longer extant. But the 
publication of these texts would at first bring little result - unlike 
the Scholastic case, since an extensive philological formation would 
be required before they could be read in the originals, while people 
so equipped have commonly not studied systematic logic. We are 
even worse off for translations than for editions; only very few 
texts (50.17) have been completely translated. Of others we have 
only fragments in western languages, in many cases nothing at all. 

On the other hand we already have a number of scientific com- 
pendia and a comprehensive history of the literature of Indian 
logic by S. C. Vidyabhusana (1921). But this is similar to Prantl's 
work in that an understanding of logical doctrine is not to be looked 
for (though it is very important in other respects), and then again 
many remarks on the literary history need revision. 

Monographs fall into two groups. One derives from the work of 
indologs, whose logical formation stems from the so-called 'classical' 
logic. The most important works are those of A. B. Keith (1921), 
H. N. Randle (1930) and Th. Shcherbatskoy (1932). Useful as they 
are, they yet contain (especially the monumental work of Shcher- 
batskoy on Buddhist logic) so many misunderstandings of systematic 
questions, that their results need to be thoroughly revised. The 
second group is composed of a few writings by indologs formed in 
mathematical logic, e.g. the works of St. Schayer (1932-3), the 
commentary of A. Kunst (1939) - both belonging to the school of 
Lukasiewicz - and the monograph of D. Ingalls (1951) which is 
perhaps the most important in the field. 

This is all very sad. Even on points of purely literary history 
there is great uncertainty. In Indological studies one is accustomed 
indeed to find the dating of a thinker fluctuating between two 
centuries, but this prejudices the possibility of solving various 
problems and important questions, e.g. the as yet obscure relation 
between Dignaga and Prasastapada, while even so prominent a 

419 



INDIAN VARIETY OF LOGIC 

logician as Vacaspati Misra can be ascribed to the 9th or 10th cen- 
tury indifferently. The content too of Indian logic is in great part 
unexplored. Ingalls' work revealed to the historian of logic a new 
intellectual world, that of the Navya-Nyaya, of which, at least in 
the west, very little was known. The mass of logical problems 
touched on there is so great that a generation of well-qualified 
investigators is needed to clear it up thoroughly. The same can be 
said of the classical period. 

To summarize, although much is still obscure, or even unknown, 
we have a certain insight into the development of Indian logic at 
the beginning of the classical period, and are even perhaps already 
in a state to understand in some degree the rise of definitely formal 
logic. Further we have some knowledge of what may be called the 
final form of this logic in the Navya-Nyaya. But that is all. We 
cannot as yet speak of a history of logical problems in India. 

D. METHOD 

In spite of this unsatisfactory state of research it seems indispens- 
able to give a brief exposition of some Indian problems, primarily 
those concerning the rise of formal logic. For, defective as is our 
knowledge of other aspects, this one can actually be followed better 
in India than in Greece. The evolution that we here describe in a 
way parallel to the other case, took in fact very much longer in 
India, so that the growth of a logical problematic can be seen in 
much more detail than in the west. 

It seems useful to complete our exposition with some details of 
later logic, presenting only fragments which have been in essentials 
taken over from Ingalls. Those doctrines are stressed which either 
illustrate the specific character of Indian logic or may be of interest 
from the standpoint of systematic logic. Of the many details not 
touched on, we may mention the highly developed sophistic. 

That our brief survey, entirely dependent on translations, is 
after all very unsatisfactory, should be evident a priori. Thanks to 
the help of competent indologs who are also logicians we yet hope 
that we have given the essentials. For the rest, it seemed better in 
the context of this book to give an incomplete exposition of Indian 
logic than to omit it entirely, for it, and it alone, serves the historian 
in the most important task of making a comparison. 

Minor alterations made to the translations we have used, are 
not noted. Additions made by the original translators and by 
ourselves are alike put in parentheses. 



420 



§ 51. THE PRECURSORS 



A. MILINDA-PANIIA 

In order to give some impression of the spirit in which the dis- 
cussions were conducted which gave rise to Indian logic, we begin 
with a passage from the Buddhist work Milinda-panha. It relates 
a conversation between the Greek king Menander, who ruled over 
the Punjab and part of what later became the United Provinces 
about 150 B.C., and the sage Nagasena. The work itself dates from 
much later. Nagasena's words reveal a world of discussion not 
unlike that which we meet in Plato. 

51.01 The king said: 'Excellent Nagasena, would you like 
to hold further discussion with me?' 

'If you are willing to discuss like a wise man, king, yes, 
indeed; but if you want to discuss like a king, then no.' 
'How do the wise discuss, excellent Nagasena?' 
'In the discussions of the wise, king, there is found 
unrolling and rolling up, convincing and conceding; agree- 
ments and disagreements are reached. And in all that, the 
wise suffer no disturbance. Thus it is, king, that the wise 
discuss.' 

b. kathAvatthu 

We can see how such a discussion was conducted, and the strictly 
defined rules that guided it, from another Buddhist work, the 
Kathdvatthu, perhaps contemporaneous with the last text. Here 
is a discussion between two disputants about the knowability of 
the human soul. 

51.02 Anuloma (The way forward') 

Theravddin: Is the soul known in the sense of real and 
ultimate fact? 

Puggalavddin: Yes. 

Theravddin: Is the soul known in the same way as a real 
and ultimate fact is known ? 

Puggalavddin: Nay, that cannot truly be said. 

Theravddin: Acknowledge your refutation: 

(1) If the soul be known in the sense of a real and ultimate 
fact, then indeed, good sir, you should also say, the soul is 
known in the same way as [any other] real and ultimate fact 
[is known]. 

(2) That which you say here is wrong, namely, (a) that we 

421 



INDIAN VARIETY OF LOGIC 

ought to say, 'the soul is known in the sense of a real and 
ultimate fact', but (b) we ought not to say, the soul is known 
in the same way as [any other] real and ultimate fact [is 
known]. 

(3) If the latter statement (b) cannot be admitted, then 
indeed the former statement (a) should not be admitted. 

(4) In affirming the former statement (a), while 

(5) denying the latter (6), you are wrong. 
51.03 Patikamma ('The way back') 

Puggalavddin: Is the soul not known in the sense of a 
real and ultimate fact? 

Theravddin: No, it is not known. 

Puggalavddin: Is it unknown in the same way as any real 
and ultimate fact is [known] ? 

Theravddin: Nay, that cannot truly be said. 

Puggalavddin: Acknowledge the rejoinder: 

(1) If the soul be not known in the sense of a real and 
ultimate fact, then indeed, good sir, you should also say: 
not known in the same way as any real and ultimate fact is 
known. 

(2) That which you say here is wrong, namely, that (a) 
we ought to say 'the soul is not known in the sense of a real 
and ultimate fact', and (b) we ought not to say: 'not known 
in the same way as any real and ultimate fact is known'. 

If the latter statement (b) cannot be admitted, then indeed 
the former statement (a) should not be admitted either. 
In affirming (b), while denying (a), you are wrong. 

Anuloma and Patikamma are only two of the five phases of the 
'first refutation' (pathama niggaha). On this first, there follow the 
second, third, fourth and fifth, differing only in small details such as 
'everywhere', 'always' and 'in everything'; then come four more in 
which 'known' and 'unknown' replace one another. 

It can hardly be denied that this procedure, which evidently 
follows a fixed rule of discussion, seems to us rather too long and 
complicated. But it is hard to understand how Randle (51.04) 
could conclude from it that the author of the Kathavatlhu had no 
respect for logic. For our text shows clearly that the disputants 
not only apply definite and accepted rules of formal logic, but almost 
formulate them expressly. 

St. Schayer also saw this (51.05). But in speaking of 'anticipations 
of propositional logic' in the Kathavatlhu, he seems to go to far. 
One could indeed think of the various statements as substitutions 
in the following propositional formulae : 

422 



PRECURSORS 

51.021 (1) If p, then q; 
therefore (2) not: p and not q; 
therefore (3) if not q, then not p. 

51.031 (1) If not p, then not q; 
therefore (2) not: not p and not not q; 
therefore (3) if not not q, then not not p. 

But that would be to credit the Indian thinkers with a power of 
abstraction which they possessed no more than the pre-Arisiotelians. 
The rules applied by our author are rather to be interpreted like 
those we found among the early Greeks (cf. § 7, D), rules, then, which 
rather correspond to the following formulae of the logic of terms: 

51.022 (1) If A is B, then A is C; 
therefore (2) not: (A is B) and not (A is C); 
therefore (3) if not (A is C), then not (A is B). 

51.032 (1) If A is not B, then A is not C; 
therefore (2) not: (A is not B) and not [A is not C); 
therefore (3) If not (A is not C), then not (A is not 

B). 
Here we note that 51.03 results from 51.02 on substitution of 'A 
is not B' for 'A is B' and of l A is not C for l A is C\ which might 
suggest that some propositional rules were already in conscious 
use. But abstract formulation of such rules is nowhere to be found 
in this context, and the substitutions show that the thought revolved 
round the fixed subject A. 

It should further be noticed that the passage from (1) to (2) 
involves a term-logical analogue of a well-known definition of 
implication (cf. 31.13; 49.04 [1.03]), and that (2) and (3) together 
constitute a kind of law of contraposition (31.20, 43.22 [28]) or 
rather a modus iollendo iollens in a term-logical version (cf. 16.16). 

The important point historically, is that the beginning of Indian 
logic corresponds so closely to that of Greek logic. 

C. THE TEN-MEMBERED FORMULA 

It may be that the Kathdvatthu exhibits a level to which the 
methodology of discussion did not often attain at that time; cer- 
tainly we find many later texts which are further removed from 
formal logic. An extract from the Dasavaikdlika-niryukti of the 
younger Bhadrabahu, a Jin, may serve as an example. He lived 
before 500 a.d., perhaps about 375 (51.06). The importance of this 
text is that it shows one a process from which the later five-membered 
syllogism may have developed. 

51.07 (1) The proposition (pratijna) : 'to refrain from taking 
life is the greatest of virtues.' 

423 



INDIAN VARIETY OF LOGIC 

(2) The limitation of the proposition (pratijna-vibhakti) : 
'to refrain from taking life is the greatest of virtues, according 
to the jinist Tirthahkaras.' 

(3) The reason (heiu) : 'to refrain from taking life is the 
greatest of virtues, because those who so refrain are loved by 
the gods, and to do them honour is an act of merit for men.' 

(4) The limitation of the reason (hetu-vibhakti): 'none but 
those who refrain from taking life are allowed to remain in 
the highest place of virtue.' 

(5) The counter-proposition (vipaksa) : 'but those who 
despise the jinist Tlrthankaras and take life are said to be 
loved by the gods, and men regard doing them honour as an 
act of merit. Again, those who take life in sacrifices are said 
to be residing in the highest place of virtue. Men, for instance, 
salute their fathers-in-law as an act of virtue, even though 
the latter despise the jinist Tlrthankaras and habitually 
take life. Moreover, those who perform animal sacrifices are 
said to be beloved of the gods.' 

(6) The opposition to the counter-proposition (vipaksa- 
pralisedha) : 'those who take life as forbidden by the jinist 
Tlrthankaras do not deserve honour, and they are certainly not 
loved by the gods. It is as likely that fire will be cold as that 
they will be loved by the gods, or to do them honour will 
be regarded by men as an act of merit. Buddha, Kapila and 
others, though really not fit to be worshipped, were honoured 
for their miraculous sayings, but the jinist Tlrthankaras 
are honoured because they speak absolute truth'. 

(7) An instance or example (drstanta) : 'the Arhats and 
Sddhus do not even cook food, lest in so doing they should take 
life. They depend on householders for their meals.' 

(8) Questioning the validity of the instance or example 
(dsankd) : 'the food which the householders cook is as much 
for the Arhats and Sddhus as for themselves. If, therefore, 
any insects are destroyed in the fire, the Arhats and Sddhus 
must share in the householders' sin. Thus the instance cited 
is not convincing.' 

(9) The meeting of the question (dsankdpratisedha) : 'the 
Arhats and Sddhus go to the householders for their food 
without giving notice, and not at fixed hours. How, therefore, 
can it be said that the householders cooked food for the 
Arhats and Sddhus? Thus the sin, if any, is not shared by the 
Arhats and Sddhus.' 

424 



SOTRAS 

(10) Conclusion (nigamana) : 'to refrain from taking life 
is therefore the best of virtues, for those who so refrain are 
loved by the gods, and to do them honour is an act of merit 
for men.' 



§ 52. VAlSESIKA- AND. NYAYA-SOTRA 

So far we have spoken of the precursors of Indian logic; now we 
shall consider the first step in its development, which takes place 
mainly in the two sister sutras, the Vaisesika- and Nydya-sutra. 
The V aisesika-sulra is older, and in most respects more important 
for logic; but the Nydya-sutra underlies the whole later development 
of Indian logic, constituting, indeed, its Organon. 

First we give the doctrine of categories from the V aisesika-sulra, 
then some short passages from the same source about inference, 
and finally go on to the Nydya-sutra and its five-membered syllogism. 

A. VAI§ESIKA-S0TRA 
1. Doctrine of categories 

52.01 The supreme good (results) from the knowledge, 
produced by a particular piety, of the essence of the predica- 
tes, substance, attribute, action, genus, species, and com- 
bination, by means of their resemblances and differences. 

Earth, water, fire, air, ether, time, space, self, and mind 
(are) the only substances. 

Attributes are colour, taste, smell, and touch, numbers, 
measures, separateness, conjunction and disjunction, priority 
and posteriority, understandings, pleasure and pain, desire 
and aversion, and volitions. 

Throwing upwards, throwing downwards, contraction, 
expansion and motion are actions. 

The resemblance of substance, attribute, and action lies 
in this that they are existent and non-eternal, have substance 
as their combinative cause, are effect as well as cause, and 
are both genus and species. 

The resemblance of substance and attribute is the charac- 
teristic of being the originators of their congeners. 

52.02 Substance-ness, attribute-ness, and action-ness are 
both genera and species. 

(The statement of genera and species has been made) with 
the exception of the final species. 

425 



INDIAN VARIETY OF LOGIC 

Existence is that to which are due the belief and usage, 
names '(It is) existent', in respect of substance, attribute 
and action. 

Existence is a different object from substance, attribute 
and action. 

And as it exists in attributes and actions, therefore it is 
neither attribute nor action. 

(Existence is different from substance, attribute and 
action) also by reason of the absence of genus-species in it. 

2. Inference 

Besides the doctrine of categories, the Vaisesika-sutra contains 
the first Indian account of inference known to us. 

52.03 'It is the effect or cause of, conjunct with, contra- 
dictory to, or combined in, this' - such is (cognition) produced 
by the mark of inference. 

'It is its' (- this cognition is sufficient to cause an illation 
to be made) ; whereas (the introduction of) the relation of 
effect and cause arises from a (particular) member (of the 
argument). 

Hereby verbal (cognition is) explained. 

Reason, description, mark, proof, instrument - these are 
not antonyms. 

(Comparison, presumption, sub-sumption, privation, and 
tradition are all included in inference by marks), because 
they depend, for their origin, upon the cognition, namely, 
'It is its'. 

B. NYAYA-SUTRA 

1. Text 

As has been said, the Nydya-siitra, the 'logical' sutra, constitutes 
the fundamental text for the whole of Indian logic. We cite some 
passages in the translation of Vidyabhusana (52.04), for their 
pioneer character: 

52.05 1. Supreme felicity is attained by the knowledge 
about the true nature of sixteen categories, viz. means of 
right knowledge, object of right knowledge, doubt, purpose, 
familiar instance, established tenet, members, confutation, 
ascertainment, discussion, wrangling, cavil, fallacy, quibble, 
futility, and occasion for rebuke. 

2. Pain, birth, activity, faults and misapprehension - on 

426 



SOTRAS 

the successive annihilation of these in the reverse order, there 
follows release. 

3. Perception, inference, comparison and word (verbal 
testimony) - these are the means of right knowledge. 

4. Perception is that knowledge which arises from the 
contact of a sense with its object and which is determinate, 
unnameable and non-erratic. 

5. Inference is knowledge which is preceded by perception, 
and is of three kinds, viz., a priori, a posteriori and 'commonly 
seen'. 

6. Comparison is the knowledge of a thing through its 
similarity to another thing previously well known. 

7. Word (verbal testimony) is the instructive assertion of 
a reliable person. 

8. It is of two kinds, viz., that which refers to matter which 
is seen and that which refers to matter which is not seen. 

52.06 25. A familiar instance is the thing about which an 
ordinary man and an expert entertain the same opinion. 

26. An established tenet is a dogma resting on the authority 
of a certain school, hypothesis, or implication. 

27. The tenet is of four kinds owing to the distinction be- 
tween a dogma of all the schools, a dogma peculiar to some 
school, a hypothetical dogma, and an implied dogma. 

32. The members (of a syllogism) are proposition, reason, 
example, application, and conclusion. 

33. A proposition is the declaration of what is to be 
established. 

34. The reason is the means for establishing what is to be 
established, through the homogeneous or affirmative character 
of the example. 

35. Likewise through heterogeneous or negative character. 

36. A homogeneous (or affirmative) example is a familiar 
instance which is known to possess the property to be estab- 
lished and which implies that this property is invariably 
contained in the reason given. 

37. A heterogeneous (or negative) example is a familiar 
instance which is known to be devoid of the property to be 
established and which implies that the absence of this property 
is invariably rejected in the reason given. 

38. Application is a winding up, with reference to the 
example, of what is to be established as being so or not 
so. 

427 



INDIAN VARIETY OF LOGIC 

39. Conclusion is the re-stating of the proposition after the 
reason has been mentioned. 

The part of this text that we find most interesting is that con- 
taining sutras 32-39, which give what is, so far as we know, the 
first description of the five-membered syllogism. The classic and 
constantly repeated example - as standard as the western 'all 
men are mortal, Socrates is a man, etc' - is the following: 

Proposition: There is fire on the mountain; 

Reason: Because there is smoke on the mountain; 

Example: As in a kitchen - not as in a lake; 

Application: It is so; 

Conclusion: Therefore it is so. 
Before attempting to comment on this formula, we should like 
to listen to the first commentator on the Nyaya-sutra. 

2. Vdtsy ay ana's commentary 

We cite, in the version of Jha (52.08), Vatsyayana's remarks on 
some of the 'members' : 

52.09 The 'statement of the proposition' is that assertion 
which speaks of the subject which is intended to be qualified 
by the property which has to be made known or proved. 

52.10 That which 'demonstrates' - i.e. makes known or 
proves - the probandum - i.e. the property to be proved (as 
belonging to the subject), - through a property common to 
the example, is the 'statement of the probans'. That is to say, 
when one notices a certain property in the subject (with regard 
to which the conclusion is to be demonstrated) and notices the 
same property also in the example, and then puts forward 
that property as demonstrating (or proving) the probandum, - 
this putting forward of the said property constitutes the 
'statement of the probans'. As an example (in connection with 
the conclusion 'sound is not eternal') we have the statement 
'because sound has the character of being a product' ; as a 
matter of fact everything that is a product, is not eternal. 

On this text the translator, Jha, remarks: 'The term sadhya is 
used in the present text rather promiscuously. It stands for the 
probandum, the predicate of the conclusion, and also for the subject, 
the thing in regard to which that character is to be demonstrated.' 

52.11 The 'statement of the probans' is that also which 
demonstrates the probandum through dissimilarity to the 

428 



SOTRAS 

example (i.e. through a property that belongs to the example 
and not to the probandum). 'How?' For example, 'sound ig 
non-eternal, because it has the character of being produced, - 
that which has not the character of being produced is always 
eternal'. 

52.12 For instance in the reasoning 'sound is non-eternal, 
because it has the character of being produced', what the 
probans 'being produced' means is that being produced, it 
ceases to be, - i.e. loses itself. - i.e. is destroyed. Here we find 
that being produced is meant to be the means of proving 
(i.e. the probans) and being non-eternal is what is proved (the 
probandum) ; and the notion that there is the relation of 
means and end between the two properties can arise only 
when the two are found to co-exist in any one thing; and it 
arises only by reason of 'similarity' (of a number of things in 
every one of which the two properties are found to coexist). 
So that when one has perceived the said relation in the familiar 
instance, he naturally infers the same in the sound also; the 
form of the inference being: 'Sound also is non-eternal, 
because it has the character of being produced, just like such 
things as the dish, the cup and the like. And this is called 
'statement of the example' (uddharana) , because it is what is 
the means of establishing, between the two properties, of the 
relation of means and end. 

53.13 When the example cited is the homogeneous one, 
which is similar to the subject, e.g. when the dish is cited as 
the example to show that it is a product and is non-eternal, we 
have the 're-affirmation' or 'application' stated in the form 
'sound is so', i.e. 'sound is a product; where the character of 
being a product is applied to the subject sound. 

When the example cited is the heterogeneous one, which is 
dissimilar to the subject, e.g. when the soul is cited as an 
example of the substance which, not being a product, is 
eternal, - the 're-affirmation' of 'application' is stated in the 
form 'sound is not so', where the character of being a product 
is reasserted of the subject sound through the denial of the 
application of the character of not being produced. Thus there 
are two kinds of re-affirmation, based upon the two kinds of 
example. 

3. Interpretation 

Combining these explanations, we get the following scheme: 

429 



INDIAN VARIETY OF LOGIC 

( 1 ) We want to prove a property - not being eternal - of a subject- 
sound. This purpose is expressed in the 'proposition'. 

(2) To effect it, we use the 'reason', which consists in another 
property - being produced - that we have noticed in sound. 

(3) We next exemplify the matter in, say, a dish, which is produced 
and is not eternal, showing that these two properties co-exist in the 
dish and other things of that kind. This is the 'example'. It can also 
be formulated negatively, as when we adduce something in which 
absence of the probandum accompanies absence of the reason; in 
the classic example this is a lake. 

(4) Having done that, we assert that the same connection between 
being produced and not being eternal occurs in the subject - sound. 
This is the 'application'. 

(5) And so we conclude that in sound too, not being eternal 
must occur. 

The reader accustomed to western logic may find this process 
strange, but the Indian formula loses its strangeness, and even 
seems quite natural, when it is remembered that it is not the result 
of reflexion about the Platonic Biadpeait;, but merely the fixing of 
a method of discussion. The following sequence is quite natural in 
discussion: 

A.: I state that S has the character P (1) 

B.: Why? 

A.: Because S has the character M (2). 

B.: So what? 

A.: Well, both M and P characterize X, and neither of them Y 
(3). So it is in our case (4). Therefore S has the character P (5). 
That is just the form of our 'syllogism'. But what logical formula 
underlies it? That question formed the subject of a centuries-long 
discussion in India, of which we have only partial knowledge and 
understanding. Some details are given in the next chapter. One 
point already emerges from the Nydya-sulra and Vatsyayana, that 
we should not look for any universal premisses, not, therefore, for a 
syllogism of the western kind. Vatsyayana does once say 'all' 
(52.10) ; but this should be regarded as accidental, for there is nothing 
of the kind in the Nydya-vdrttika uf Uddyotakara. Later history also 
shows how difficult the Indian logicians found it to grasp the uni- 
versal. The original formula of the sutras is simply an argument by 
analogy from some individuals to others, rhetorical rather than 
logical in character. Neither the sutras nor Vatsyayana have 
achieved a properly formal logic. 

It has been objected to this interpretation of the formula that 
besides the 'syllogism' the sutras give another means of knowledge, 
the 'comparison' (cf. 52.05, 3 to 7). But this is again an argument by 
analogy, so that we should have to accept two such arguments in 
the sutras. However, the objection is not sound, since the 'comparison 

430 



FORMAL LOGIC 

(upamdna) was expounded in the Nyaya tradition, not, as ail argu- 
ment from analogy in the ordinary sense, but as one of a quite 
special (metalinguistic) kind, an argument about a name. This 
can best be seen in the Tarka-Samgraha, a late text, but true to 
the Nyaya tradition: 

52.14 Comparison (upamdna) is the efficient cause of 
knowledge of similarity. This (in its turn) is knowledge of the 
relation between a name and the thing it names. . . . Example : 
Whoso does not know the gayal, hears from some forester 
that it is like the domestic ox ; going into the forest and remem- 
bering this saying, he sees an object like a domestic ox. At 
once there arises in him knowledge by similarity: That is 
what is called a "gayal".' 

So, following the plain text of the sutra and Vatsyayana, we 
may take the pretended 'syllogism' not as a syllogism but as a 
formula for inference by analogy, of a rhetorical kind. We shall now 
see how this became a genuine law, or rule, of formal logic. 



§53.THE RISE OF FORMAL LOGIC 

A. MAIN STAGES OF DEVELOPMENT 

What Plato is to Aristotle in formal logic, the Nyaya-sutra is to, 
say, Dharmaklrti, save that the Nyaya-sutra lacks that idea of 
universal law which in Plato opened the way to the rise of western 
logic. It was this idea which brought about the speedy emergence of 
logical form in the west. But in India, logic took shape very slowly, 
in the course of centuries and under the auspices of methodology. 
However, it is just this step-by-step, 'natural' development of 
Indian logic which gives it its great historical interest. 

Though only partially acqainted with this development, we can 
determine some of its phases. Their order of succession is not 
altogether clear, but we can be certain of their occurrence and 
sometimes of their temporal relationship. Thus we obtain the 
following scheme: 

First step: The establishment of a formal rule of syllogism (the 
trairupya) based on examples. According to G. Tucci (53.01) Vasu- 
bandhu will have known this. 

Second step: Dignaga developed the trairupya into a formal 
syllogistic - the 'wheel of reasons' (hetu-cakra). Uddyotakara carried 
this further still. 

Third step: The components of the syllogism were reduced to 
three, the probandum no longer being resumed in the conclusion, 

431 



INDIAN VARIETY OF LOGIC 

and the application being dropped. A distinct